Models and integral differentials of hyperelliptic curves

Abstract

Let $C\; : \;y^2=f(x)$ be a hyperelliptic curve of genus $g\geq 1$, defined over a complete discretely valued field $K$, with ring of integers $O_K$. Under certain conditions on $C$, mild when residue characteristic is not $2$, we explicitly construct the minimal regular model with normal crossings $\mathcal{C}/O_K$ of $C$. In the same setting we determine a basis of integral differentials of $C$, that is an $O_K$-basis for the global sections of the relative dualising sheaf $\omega _{\mathcal{C}/O_K}$.

1. Introduction

The purpose of this paper is to construct regular models of hyperelliptic curves and to describe a basis of integral differentials attached to them. Moreover, we want these constructions explicit and easy to compute.

1.1. Overview

To describe the arithmetic of curves over global fields, for example in the study of the Birch & Swinnerton-Dyer conjecture, it is essential to understand regular models and integral differentials over all primes, including those with very bad reduction. Constructing regular models of curves over discrete valuation rings is not an easy problem, even in the hyperelliptic curve case. In fact, there is no practical algorithm able to determine a model, unless the genus of the curve is $1$ or we have some tameness or nondegeneracy hypothesis.

One possible approach to tackle this problem is giving a full classification of possible regular models in a fixed genus, as done by the Kodaira–Néron [7, 19] and Namikawa–Ueno [10, 18] classifications for curves of genera $1$ and $2$ , respectively. However, this strategy seems impractical in general, since the number of models grows fast with the genus. Recently, new approaches based on clusters [14], Newton polytopes [1], and MacLane valuations [21], have been developed (see Section 1.5 for more detail).

On one side, clusters define nice and clear invariants from which one can extract information on the local arithmetic of hyperelliptic curves. Such invariants turn out to be particularly useful from a Galois theoretical point of view. However, for describing regular models, restrictions on the reduction type of the curve and on the residue characteristic of its base field [5, 14] need to be imposed. On the other side, Newton polytopes and MacLane valuations have a potential to solve the problem in general, but the respective constructions are more algorithmic and so do not give the result in closed form. Furthermore, they often depend on the chosen equation rather than on the curve itself.

In this paper, we present a new approach that preserves both positive aspects from the above and provides a link between the two sides. We describe a model from simple invariants defined from what we call rational cluster picture (Definition 1.10). This object modifies the theory in [14] and appears to be more suitable for our purpose (see Section 1.3). In fact, the rational cluster picture also carries intrinsic connections with the other presented approaches, as it is closely related to Newton polygons and to degree $1$ MacLane valuations (see [3]). When these valuations are enough to describe a regular model we say that the curve has an almost rational cluster picture (Definition 1.1; see also Corollary 3.29, Proposition 3.31). It turns out that the approach even works in residue characteristic 2, under an extra assumption that the curve is $y$ -regular (Definition 1.4). Our main result is:

Let $K$ be a complete 1 discretely valued field with $\textrm{char}(K)\neq 2$ , and let $K^{nr}$ be its maximal unramified extension. Let $C/K$ be a hyperelliptic curve, having an almost rational cluster picture over $K^{nr}$ . If the residue characteristic of $K$ is $2$ , assume that $C_{K^{nr}}$ is $y$ -regular. Then via the rational cluster picture we determine:

1. (i) the minimal regular model with normal crossings $\mathcal{C}^{\textrm{min}}$ ,

2. (ii) a basis of integral differentials of $C$ .

This result applies to a wide class of curves, covering wild cases and base fields with even residue characteristic. For example, if $g=2$ , then $107$ out of $120$ Namikawa-Ueno types [18] arise from hyperelliptic curves satisfying the conditions of our theorem. In addition, the author believes it has a potential to solve the problem in general. Heuristically speaking, the rational clusters invariants are expected to extend to general MacLane valuations. This approach could eventually lead to a full characterisation of minimal models with normal crossings of hyperelliptic curves (over any discretely valued field).

1.2. Main results

We will now present (a simplified version of) the main results of this paper. We will then illustrate them with an explicit example in Section 1.4.

Let $K$ be a complete discretely valued field of residue characteristic $p$ , with normalised discrete valuation $v$ and ring of integers $O_K$ . We require $\textrm{char}(K)$ to be not $2$ , but we allow $p=2$ and $p=0$ . In this subsection we will assume for simplicity that $K=K^{nr}$ . Extend the valuation $v$ to an algebraic closure $\bar{K}$ of $K$ . Let $C/K$ be a hyperelliptic curve, that is a geometrically connected smooth projective curve, double cover of ${\mathbb{P}}^1_K$ . Let $g$ be the genus of $C$ . Assume $g\geq 1$ . Fix a Weierstrass equation

\begin{align*} C\;:\;y^2=f(x). \end{align*}

Let $\mathfrak{R}$ be the set of roots of $f$ in $\bar{K}$ . Thus

\begin{align*} f(x)=c_f\prod _{r\in \mathfrak{R}}(x-r). \end{align*}

For any $r,r'\in \mathfrak{R}$ , with $r\neq r'$ , denote by $\mathcal{D}_{r,r^{\prime}}$ the smallest $v$ -adic disc containing $r$ and $r'$ .

Definition 1.1 (Definition 3.26). We say that $C$ has an almost rational cluster picture if for any roots $r,r'\in \mathfrak{R}$ with $r\neq r'$ , either

1. (a) $\mathcal{D}_{r,r^{\prime}}\cap K\neq \varnothing$ , or

2. (b) $p\gt 0$ and $|\mathcal{D}_{r,r^{\prime}}\cap \mathfrak{R}|\leq |v(r-w)|_p$ for some $w\in K$ ,

where $|\cdot |_p$ denotes the canonical $p$ -adic absolute value on $\mathbb{Q}$ .

Definition 1.2. A rational cluster is a non-empty subset $\mathfrak{s}\subset \mathfrak{R}$ of the form $\mathcal{D}\cap \mathfrak{R}$ , where $\mathcal{D}$ is a $v$ -adic disc $\mathcal{D}=\{x\in \bar{K}\mid v(x-w)\geq \rho \}$ for some $w\in K$ and $\rho \in \mathbb{Q}$ . We denote by $\Sigma _K$ the set of rational clusters.

In the following definition we introduce most of the notation and quantities, associated with rational clusters, needed in order to state our main theorems.

Definition 1.3. For any $\mathfrak{s}\in \Sigma _K$ we say:

$\mathfrak{s}$ proper,	$|\mathfrak{s}|\gt 1$
$\mathfrak{s}^{\prime}$ is a child of $\mathfrak{s}$ ,	$\mathfrak{s}^{\prime}\in \Sigma _K$ and $\mathfrak{s}^{\prime}\subsetneq \mathfrak{s}$ is a maximal subcluster
$\mathfrak{s}$ minimal,	$\mathfrak{s}$ has no proper children
$\mathfrak{s}$ übereven,	$\mathfrak{s}=\bigcup _{\mathfrak{s}^{\prime}\text{ child of }\mathfrak{s}}\mathfrak{s}^{\prime}$ and $|\mathfrak{s}^{\prime}|$ even for all children $\mathfrak{s}^{\prime}$ of $\mathfrak{s}$

Moreover, we write $\mathfrak{s}^{\prime}\lt \mathfrak{s}$ , or $\mathfrak{s}=P(\mathfrak{s}^{\prime})$ , for a child $\mathfrak{s}^{\prime}\in \Sigma _K$ of $\mathfrak{s}$ , and $r\wedge \mathfrak{s}$ for the smallest rational cluster containing the root $r\in \mathfrak{R}$ and $\mathfrak{s}$ .

Let $\Sigma _K$ be the set of proper rational clusters. For any $\mathfrak{s}\in \Sigma _K$ , define its radius

\begin{align*} \rho _{\mathfrak{s}}=\max _{w\in K}\min _{r\in \mathfrak{s}} v(r-w) \end{align*}

and the following quantities:

$b_{\mathfrak{s}}$	denominator of $\rho_{\mathfrak{s}}$
$\epsilon _{\mathfrak{s}}$	$v(c_f) + \sum _{r\in \mathfrak{R}} \rho _{r\wedge \mathfrak{s}}$
$D_{\mathfrak{s}}$	$1$ if $b_{\mathfrak{s}}\epsilon _{\mathfrak{s}}$ odd, $2$ if $b_{\mathfrak{s}}\epsilon _{\mathfrak{s}}$ even
$m_{\mathfrak{s}}$	$(3-D_{\mathfrak{s}})b_{\mathfrak{s}}$
$p_{\mathfrak{s}}$	$1$ if $|\mathfrak{s}|$ is odd, $2$ if $|\mathfrak{s}|$ is even
$s_{\mathfrak{s}}$	$\frac 12(|\mathfrak{s}|\rho _{\mathfrak{s}}+p_{\mathfrak{s}}\rho _{\mathfrak{s}}-\epsilon _{\mathfrak{s}})$
$\gamma _{\mathfrak{s}}$	$2$ if $|\mathfrak{s}|$ is even and $\epsilon _{\mathfrak{s}}\!-\!|\mathfrak{s}|\rho _{\mathfrak{s}}$ is odd, $1$ otherwise
$p_{\mathfrak{s}}^0$	$1$ if $\mathfrak{s}$ is minimal and $\mathfrak{s}\cap K\neq \varnothing$ , $2$ otherwise
$s_{\mathfrak{s}}^0$	$-\epsilon _{\mathfrak{s}}/2+\rho _{\mathfrak{s}}$
$\gamma _{\mathfrak{s}}^0$	2 if $p_{\mathfrak{s}}^0=2$ and $\epsilon _{\mathfrak{s}}$ is odd, 1 otherwise

Definition 1.4 (Definition 4.10). We say that the hyperelliptic curve $C$ is $y$ -regular if either $p\neq 2$ or $D_{\mathfrak{s}}=1$ for any $\mathfrak{s}\in \Sigma _K$ .

Definition 1.5. Let $\mathfrak{s}\in \Sigma _K$ and let $c\in \{0,\dots,b_{\mathfrak{s}}-1\}$ such that $c\rho _{\mathfrak{s}}-\frac{1}{b_{\mathfrak{s}}}\in \mathbb{Z}$ . Define

\begin{align*} \tilde{\mathfrak{s}}=\{\mathfrak{s}^{\prime}\in \Sigma _K\cup \{\varnothing \}\mid \mathfrak{s}^{\prime}\lt \mathfrak{s}\text{ and }\tfrac{|\mathfrak{s}^{\prime}|}{b_{\mathfrak{s}}}-c\epsilon _{\mathfrak{s}}\notin 2\mathbb{Z}\}, \end{align*}

where $\varnothing \lt \mathfrak{s}$ if $\mathfrak{s}$ is minimal and $p_{\mathfrak{s}}^0=2$ .

The genus $g(\mathfrak{s})$ of a rational cluster $\mathfrak{s}\in \Sigma _K$ is defined as follows:

• If $D_{\mathfrak{s}}=1$ , then $g(\mathfrak{s})=0$ .

• If $D_{\mathfrak{s}}=2$ , then $2g(\mathfrak{s})+1$ or $2g(\mathfrak{s})+2$ equals $\dfrac{|\mathfrak{s}|-\sum _{\mathfrak{s}^{\prime}\lt \mathfrak{s}}|\mathfrak{s}^{\prime}|}{b_{\mathfrak{s}}}+|\tilde{\mathfrak{s}}|$ .

Notation 1.6. Let $\alpha \in \mathbb{Z}_+$ , $a,b\in \mathbb{Q}$ , with $a\gt b$ , and fix $\frac{n_i}{d_i}\in \mathbb{Q}$ so that

\begin{align*} \alpha a=\frac{n_0}{d_0}\gt \frac{n_1}{d_1}\gt \ldots \gt \frac{n_r}{d_r}\gt \frac{n_{r+1}}{d_{r+1}}=\alpha b,\quad \text{with}\quad \begin{vmatrix}n_i & &n_{i+1}\\ d_i & &d_{i+1}\\\end{vmatrix}=1, \end{align*}

and $r$ minimal. We write ${\mathbb{P}}^1(\alpha,a,b)$ for a chain of ${\mathbb{P}}^1$ s (Notation 4.16) of length $r$ and multiplicities $\alpha d_i,\dots,\alpha d_r$ . Denote by ${\mathbb{P}}^1(\alpha,a)$ the chain ${\mathbb{P}}^1(\alpha,a,\lfloor \alpha a-1\rfloor/\alpha )$ .

The following theorem describes the special fibre of a regular model of $C$ with strict normal crossings.2 It follows from a more general result constructing a proper flat model of $C$ unconditionally (Theorem 4.18). For the special fibre $\mathcal{C}^{\textrm{min}}_s$ of the minimal regular model with normal crossings, the reader can refer to Theorem 4.23, where we also describe a defining equation for all components of $\mathcal{C}_s^{\textrm{min}}$ and discuss the Galois action (for general $K$ ). Finally, note that all these models are constructed in Section 5 by giving an explicit open affine cover (see Sections 5.1–5.3).

Theorem 1.7 (Regular SNC model). Suppose $C$ is $y$ -regular and has almost rational cluster picture. Then we can explicitly construct a regular model with strict normal crossings $\mathcal{C}/O_{K}$ of $C$ (Sections 5.1–5.3 ). Its special fibre $\mathcal{C}_s/k$ is given as follows.

1. (1) Every $\mathfrak{s}\in \Sigma _K$ gives a $1$ -dimensional closed subscheme $\Gamma _{\mathfrak{s}}$ of multiplicity $m_{\mathfrak{s}}$ . If $\mathfrak{s}$ is übereven and $\epsilon _{\mathfrak{s}}$ is even, then $\Gamma _{\mathfrak{s}}$ is the disjoint union of $\Gamma _{\mathfrak{s}}^{-}\simeq{\mathbb{P}}^1$ and $\Gamma _{\mathfrak{s}}^{+}\simeq{\mathbb{P}}^1$ , otherwise $\Gamma _{\mathfrak{s}}$ is a smooth geometrically integral curve of genus $g(\mathfrak{s})$ (write $\Gamma _{\mathfrak{s}}^{-}=\Gamma _{\mathfrak{s}}^{+}=\Gamma _{\mathfrak{s}}$ in this case).

2. (2) Every $\mathfrak{s}\in \Sigma _K$ with $D_{\mathfrak{s}}=1$ gives $(|\mathfrak{s}|-\sum _{\mathfrak{s}^{\prime}\in \Sigma _K,\,\mathfrak{s}^{\prime}\lt \mathfrak{s}}|\mathfrak{s}^{\prime}|+p_{\mathfrak{s}}^0-2)/b_{\mathfrak{s}}$ open-ended ${\mathbb{P}}^1$ s of multiplicity $b_{\mathfrak{s}}$ from $\Gamma _{\mathfrak{s}}$ .

3. (3) Finally, for any $\mathfrak{s}\in \Sigma _K$ draw the following chains of ${\mathbb{P}}^1$ s:

Conditions	Chain	From	To
$\mathfrak{s}$ minimal	${\mathbb{P}}^1(\gamma _{\mathfrak{s}}^0,-s_{\mathfrak{s}}^0)$	$\Gamma _{\mathfrak{s}}^-$	open-ended
$\mathfrak{s}$ minimal, $p_{\mathfrak{s}}^0/\gamma _{\mathfrak{s}}^0=2$	${\mathbb{P}}^1(\gamma _{\mathfrak{s}}^0,-s_{\mathfrak{s}}^0)$	$\Gamma _{\mathfrak{s}}^+$	open-ended
$\mathfrak{s}\neq \mathfrak{R}$	${\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}},s_{\mathfrak{s}}-p_{\mathfrak{s}}\cdot \frac{\rho _{\mathfrak{s}}-\rho _{P(\mathfrak{s})}}{2})$	$\Gamma _{\mathfrak{s}}^-$	$\Gamma _{P(\mathfrak{s})}^-$
$\mathfrak{s}\neq \mathfrak{R}$ , $p_{\mathfrak{s}}/\gamma _{\mathfrak{s}}=2$	${\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}},s_{\mathfrak{s}}-p_{\mathfrak{s}}\cdot \frac{\rho _{\mathfrak{s}}-\rho _{P(\mathfrak{s})}}{2})$	$\Gamma _{\mathfrak{s}}^+$	$\Gamma _{P(\mathfrak{s})}^+$
$\mathfrak{s}=\mathfrak{R}$	${\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}})$	$\Gamma _{\mathfrak{s}}^-$	open-ended
$\mathfrak{s}=\mathfrak{R}$ , $p_{\mathfrak{s}}/\gamma _{\mathfrak{s}}=2$	${\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}})$	$\Gamma _{\mathfrak{s}}^+$	open-ended

Definition 1.8. For any $\mathfrak{s}\in \Sigma _K$ , an element $w_{\mathfrak{s}}\in K$ is called rational centre of $\mathfrak{s}$ if $\min _{r\in \mathfrak{s}}v(r-w_{\mathfrak{s}})=\rho _{\mathfrak{s}}$ .

If $\mathfrak{s}^{\prime}\lt \mathfrak{s}$ and $w_{\mathfrak{s}^{\prime}}$ is a rational centre of $\mathfrak{s}^{\prime}$ , then $w_{\mathfrak{s}^{\prime}}$ is also a rational centre of $\mathfrak{s}$ . For any minimal rational cluster $\mathfrak{s}^{\prime}$ fix a rational centre $w_{\mathfrak{s}^{\prime}}$ . For any $\mathfrak{s}\in \Sigma _K$ fix $w_{\mathfrak{s}}=w_{\mathfrak{s}^{\prime}}$ for some minimal rational cluster $\mathfrak{s}^{\prime}\subseteq \mathfrak{s}$ .

The following result gives a basis of integral differentials when $K=K^{nr}$ . In Theorem 6.4 we extend it to the case $K\neq K^{nr}$ .

Theorem 1.9 (Theorem 6.3). Suppose $C$ is $y$ -regular and has almost rational cluster picture. For $i=0,\dots,g-1$ , inductively

1. (i) define $e_i\;:\!=\;\displaystyle \max _{\mathfrak{t}\in \Sigma _K}\bigg \{\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}-\sum _{j=0}^{i-1}\rho _{\mathfrak{s}_j\wedge \mathfrak{t}}\bigg \}$ ;

2. (ii) let $\Sigma _i=\displaystyle \bigg \{\mathfrak{t}\in \Sigma _K\mid \,e_i=\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}-\sum _{j=0}^{i-1}\rho _{\mathfrak{s}_j\wedge \mathfrak{t}}\bigg \}$ ;

3. (iii) choose a maximal element $\mathfrak{s}_i$ of $\Sigma _i$ freely.

Then a basis of integral differentials is given by

\begin{align*} \mu _i=\pi ^{\lfloor e_i\rfloor }\prod _{j=0}^{i-1}(x-w_{\mathfrak{s}_j})\frac{dx}{2y},\qquad i=0,\dots,g-1. \end{align*}

Note that given $e_i$ as in the previous theorem, the sum $\sum _{i=0}^{g-1}\lfloor e_i \rfloor$ is the quantity, often denoted by $v({\omega ^\circ }/{\omega })$ , appearing in the period in the Birch and Swinnerton-Dyer conjecture (for more details see [4], [25, §1.3]).

1.3. Rational cluster picture

In this subsection we define the rational cluster picture and compare it with the classical cluster picture defined in [14]. We will show, via a simple example, in which sense the new object we introduce appears to be more suitable for the study of regular models.

Definition 1.10 (Definition 3.9). Let $K$ and $C$ as before. The rational cluster picture of $C$ is the collection of its rational clusters $\Sigma _K$ together with their radii.

Example 1.11. Let $p$ be any prime number and set $K=\mathbb{Q}_p^{nr}$ . Let $E_p/\mathbb{Q}_p^{nr}$ given by $y^2=x^3-p$ . Then $E_p$ is an elliptic curve with Kodaira-Néron reduction type II. Therefore, the minimal regular model (with normal crossings) of $E_p$ does not depend on $p$ . This is in accordance with the fact that the rational cluster picture of $E_p$ is the same for all $p$ . Indeed, the set of roots of the polynomial $x^3-p$ is $\mathfrak{R}=\{\sqrt [3]{p}, \zeta _3\sqrt [3]{p}, \zeta _3^2\sqrt [3]{p}\}$ , where $\zeta _3$ is a primitive $3$ rd of unity. Hence the rational cluster picture of $E_p$ is

where we denoted with bullet points the roots in $\mathfrak{R}$ , with a surrounding oval the only rational cluster $\mathfrak{R}$ , and with the subscript the radius $\rho _{\mathfrak{R}}$ of $\mathfrak{R}$ .

A different behaviour is observed when we consider the cluster picture [14 , Definition 1.26] of $E_p$ , collection of its clusters together with their depths. The cluster picture of $E_p$ is

where the subscripts represent the depth of the cluster $\mathfrak{R}$ . It does depend on $p$ and differs from the rational cluster picture when $p=3$ . Thus, although the cluster picture is particularly useful for Galois theoretical problems, the rational cluster picture appears to be a more suitable object for the study of regular models of the curve.

Finally, note that $E_p$ has an almost rational cluster picture. For any two distinct roots $r,r'\in \mathfrak{R}$ , the smallest $v$ -adic disc $D_{r,r^{\prime}}$ containing them also contains the whole $\mathfrak{R}$ . The element $0\in \mathbb{Q}_p^{nr}$ belongs to $D_{r,r^{\prime}}$ when $p\neq 3$ , while $|D_{r,r^{\prime}}\cap \mathfrak{R}|=3=|v(r)|_p$ , if $p=3$ .

The advantages of the rational cluster picture discussed in this subsection can also be observed in the following example where we study a more complex family of hyperelliptic curves having almost rational cluster picture.

1.4. Example

In this subsection we are going to present an example of a family of hyperelliptic curves $C_p$ satisfying the hypothesis of Theorems 1.7 and 1.9. Via those results we will then describe the special fibre of the minimal regular model and a basis of integral differentials of $C_p$ . All the computations involved are explained in detail in Examples 3.32, 4.25 and 6.5.

For any prime number $p$ , let $a\in \mathbb{Z}_p$ , $b\in \mathbb{Z}_p^\times$ such that the polynomial $x^2+ax+b$ is not a square modulo $p$ . Let $C_p/\mathbb{Q}_p$ be the hyperelliptic curve of genus $4$ given by $y^2=f(x)$ , where $f(x)=(x^6+ap^4x^3+bp^8)((x-p)^3-p^{11})$ . The curve $C_p/\mathbb{Q}_p^{nr}$ has an almost rational cluster picture and is $y$ -regular when $p=2$ . Its rational cluster picture is

where $\rho _{\mathfrak{t}_3}=\frac{4}{3}$ , $\rho _{\mathfrak{t}_4}=\frac{11}{3}$ , and $\rho _{\mathfrak{R}}=1$ . From Theorem 1.7 we can construct a regular model with strict normal crossings of $C_p$ with special fibre

over $\bar{\mathbb{F}}_p$ . Computing the self-intersection of each irreducible component we easily see that this model coincides with the minimal regular model $\mathcal{C}^{\textrm{min}}$ . Theorem 4.23 also describes the action of the Galois group $\textrm{Gal}(\bar{\mathbb{F}}_p/{\mathbb{F}}_p)$ on the special fibre $\mathcal{C}_s^{\textrm{min}}$ of $\mathcal{C}^{\textrm{min}}$ . If the roots of $x^2+ax+b\mod p$ are in ${\mathbb{F}}_p$ then the absolute Galois group acts trivially on each component, otherwise it swaps the $2$ irreducible components of multiplicity $3$ intersecting $\Gamma _{\mathfrak{t}_3}$ .

From Theorem 1.9 it follows that, for any $p$ , a basis of integral differentials of $C_p/\mathbb{Q}_p^{nr}$ is given by

\begin{align*} \mu _0=p^4\cdot \frac{dx}{2y},\quad \mu _1=p^3(x-p)\cdot \frac{dx}{2y},\quad \mu _2=p(x-p)x\cdot \frac{dx}{2y},\quad \mu _3=(x-p)x^2\cdot \frac{dx}{2y}. \end{align*}

In fact, this is also a basis of integral differentials of $C_p/\mathbb{Q}_p$ since they are all defined over $\mathbb{Q}_p$ (see Proposition B.2).

Below we will present related works of other authors concerning regular models and integral differentials of hyperelliptic curves. Note that the example presented here is not covered by [14] and [1] since the curve $C_p$ is not semistable and not $\Delta _v$ -regular. In fact, if $p=3$ the curve $C_p$ does not even have tamely potential semistable reduction. The results in [5] assume $p\gt 2$ and $C_p$ with tamely potential semistable reduction, hence they cannot be used when $p=2,3$ . Finally, there is no classification for genus $4$ curves.

1.5. Related works of other authors

Let $K$ be a discretely valued field with residue field $k$ of characteristic $p$ and let $C/K$ be a hyperelliptic curve of genus $g$ .

In genus $1$ , when $k$ is perfect, thanks to Tate’s algorithm, one can describe the minimal regular model and the space of integral differentials of an elliptic curve $C$ (see e.g., [24, IV.8.2], [9, Theorem 9.4.35]).

If $K={\mathbb{C}}(t)$ and $C$ has genus $2$ , then Namikawa and Ueno [18] and Liu [12] give a full classification of the possible configurations of the special fibre of the minimal regular model of $C$ .

If $p\neq 2$ , then Liu and Lorenzini show in [13] that regular models of $C$ can be seen as double cover of well-chosen regular models of ${\mathbb{P}}^1_K$ . Since the latter can be found by using the MacLane valuations [15] approach in [21], this argument gives a way to describe any regular model of a hyperelliptic curve. At the moment there is no known closed form description of a regular model based on this approach and it has not been generalised to the $p=2$ case.

If $p\gt 2$ , $k$ finite, and $C$ is semistable, then in [14] the authors explicitly construct a minimal regular model in terms of the cluster picture of $C$ . Under the same assumptions, Kunzweiler [8] gives a basis of integral differentials rephrasing [6, Proposition 5.5] in terms of the cluster invariants introduced in [14]. These results can be recovered from Theorem 4.23 (see Corollary 4.27) and Theorem 6.3.

If $p\gt 2$ and $C$ is semistable over some tamely ramified extension $L/K$ , then Faraggi and Nowell [5] find the special fibre of the minimal regular model of $C$ with strict normal crossings taking the quotient of the stable model of $C_L$ and resolving the (tame) singularities. However, since they do not describe the charts of the model, their result does not immediately yield all arithmetic invariants, such as a basis of integral differentials.

The last work we want to recall represents an important ingredient of the strategy we will use in this paper (described more precisely in the next subsection). T. Dokchitser in [1] shows that the toric resolution of $C$ gives a regular model in case of $\Delta _v$ -regularity [1, Definition 3.9]. This result, used also in [5], holds for general curves and in any residue characteristic. In his paper, Dokchitser also describes a basis of integral differentials since his model is given as open cover of affine schemes. In Corollary 3.25 and Theorem 6.1, we will rephrase his results for hyperelliptic curves by using rational cluster picture invariants from Section 3.

1.6. Strategy and outline of the paper

In [1], Dokchitser not only describes a regular model of $C$ in case of $\Delta _v$ -regularity, but also constructs a proper flat model $\mathcal{C}_\Delta$ without any assumptions on $C$ . Assume $C$ is $y$ -regular and has an almost rational cluster picture over $K^{nr}$ with rational centres $w_1,\dots, w_m\in K^{nr}$ . Our approach to construct the minimal regular model with normal crossings of $C$ is composed by the following steps:

• Consider the $x$ -translated hyperelliptic curves $C^{w_h}/K^{nr}:y^2=f(x+w_h)$ , for $h=1,\dots,m$ . For each $h$ , [1, Theorem 3.14] constructs a proper flat model $\mathcal{C}_\Delta ^{w_h}$ , possibly singular.

• We glue regular open subschemes of these models along common opens, and show that the result is a proper flat regular model $\mathcal{C}$ of $C_{K^{nr}}$ with strict normal crossings.

• We give a complete description of what components of the special fibre of $\mathcal{C}$ have to be blown down to obtain the minimal model with normal crossings $\mathcal{C}^{\textrm{min}}$ of $C_{K^{nr}}$ .

• Finally, we describe the action of the absolute Galois group $G_k$ of $k$ on the special fibre of $\mathcal{C}^{\textrm{min}}$ .

We will explicitly describe both the models $\mathcal{C}_\Delta ^{w_h}$ and $\mathcal{C}$ . This allows us to study the global sections of its relative dualising sheaf $\omega _{\mathcal{C}/O_K}(\mathcal{C})$ .

In Section 2, we present some results on Newton polygons used in the following sections. In Section 3, we recall the basic objects and notation of [14] and define the rational cluster picture. Moreover, we relate it with the notions given in Section 2. This comparison allows us to rephrase the objects in [1] in terms of rational clusters invariants in Section 4. In the same section we also state the theorems which describe the special fibres of a proper flat model (Theorem 4.18) and of the minimal regular model with normal crossings (Theorem 4.23) of $C$ . The construction of these models, from which the two theorems above follow, is presented in Section 5. Finally, in Section 6, Theorems 6.3 and 6.4 describe a basis of integral differentials of $C$ , in terms of rational clusters invariants defined in Section 3.

1.7. Notation

In the following, we present the main notation used for fields, hyperelliptic curves and Newton polytopes.

$K,v$	complete field with normalised discrete valuation $v$
$O_k,\pi,k,p$	ring of integers, uniformiser, residue field, $\textrm{char}(k)$
$\bar{K},\bar{k}$	fixed algebraic closure of $K$ , residue field of $\bar{K}$
$K^{\textrm{s}}, K^{nr}$	separable closure, maximal unramified extension of $K$ in $\bar{K}$
$O_{K^{nr}},k^{\textrm{s}}$	ring of integers of $K^{nr}$ , residue field of $K^{nr}$
$F$	extension of $K$ in $\bar{K}$ , unramified in Section 4
$G_K, G_k$	absolute Galois groups $\textrm{Gal}(K^{\textrm{s}}/K), \textrm{Gal}(k^{\textrm{s}}/k)$
$f(x)$	$=\sum a_i x^i$ , polynomial in $K[x]$ , separable from Section 3
$\texttt{NP}(f)$	Newton polygon of $f$ , lower convex hull of $\{(i,v(a_i))\mid i\}$
$f|_L,\overline{f|_L}$	restriction and reduction of $f$ to an edge $L$ of $\texttt{NP}(f)$ (Definition 2.5)
$g(x,y)$	$=y^2-f(x)$ , polynomial in $K[x,y]$ defining $C$
$C$	hyperelliptic curve defined over $K$ by $g(x,y)=0$
$f_w(x), f_h(x)$	$=f(x+w), f(x+w_h)$ , for a given rational centre $w_h$
$g_w(x,y), g_h(x,y)$	$=y^2-f_w(x), y^2-f_h(x)$
$C^w$	$\simeq C$ , hyperelliptic curve given by $g_w(x,y)=0$
$\Delta ^w, \Delta _v^w$	Newton polytopes attached to $C^w$ as in [1, §1.1] (Notation 4.1)
$F_{\mathfrak{t}}^w,L_{\mathfrak{t}}^w,V_{\mathfrak{t}}^w,V_0^w$	$v$ -faces and $v$ -edges of $\Delta ^w$ (Notation 4.4)
$s_1^\lambda, s_2^\lambda, r_\lambda$	$s_1^\lambda, s_2^\lambda \in \mathbb{Q}$ , $r_\lambda \in \mathbb{Z}_{\geq 0}$ , attached to a $v$ -edge of $\Delta ^w$ (Notation 4.2)

For a separable polynomial $f\in k[x]$ or a hyperelliptic curve $C/K:y^2=f(x)$ as above, the following is the main notation for clusters.

$c_f, \mathfrak{R}$	leading coefficient and set of roots of $f$
$\Sigma _f,\Sigma _C$	cluster picture, the set of clusters of $f$ , $C$ (Definition 3.2)
$\mathfrak{s}\in \Sigma _C$	cluster, $\mathfrak{s}=\mathcal{D}\cap \mathfrak{R}$ , for a $v$ -adic disc $\mathcal{D}$ (Definition 3.1)
$G_{\mathfrak{s}}, K_{\mathfrak{s}}, k_{\mathfrak{s}}$	$G_{\mathfrak{s}}=\textrm{Stab}_{G_K}({\mathfrak{s}})$ ; $K_{\mathfrak{s}}=\left ( K^{\textrm{s}}\right )^{G_{\mathfrak{s}}}$ ; $k_{\mathfrak{s}}$ residue field of $K_{\mathfrak{s}}$
$d_{\mathfrak{s}}$	$=\min _{r,r^{\prime}\in \mathfrak{s}}v(r-r')$ is the depth of a cluster $\mathfrak{s}$ (Definition 3.1)
$\mathfrak{s}^{\prime}\lt \mathfrak{s}=P({\mathfrak{s}}^{\prime})$	$\mathfrak{s}^{\prime}$ is a child of $\mathfrak{s}$ and $\mathfrak{s}$ is the parent of $\mathfrak{s}^{\prime}$ (Definition 3.3)
$\mathfrak{s}\wedge \mathfrak{t}$	smallest cluster containing $\mathfrak{s}$ and $\mathfrak{t}$ (Definition 3.3)
$\rho _{\mathfrak{s}}$	$=\max _{w\in F}\min _{r\in \mathfrak{s}} v(r-w)$ , radius of $\mathfrak{s}\in \Sigma _{C_F}$ (Definitions 3.8 and 4.6)
$b_{\mathfrak{s}}$	denominator of $\rho _{\mathfrak{s}}$ (Definition 4.6)
$w_{\mathfrak{s}}$	rational centre of $\mathfrak{s}$ (Definition 3.8)
$\epsilon _{\mathfrak{s}}$	$=v(c_f) + \sum _{r\in \mathfrak{R}} \rho _{r\wedge \mathfrak{s}}$ (Definitions 3.19 and 4.6)
$\Sigma _f^{\textrm{rat}},\Sigma _C^{\textrm{rat}}$	rational cluster picture (Definition 3.9)
$\mathfrak{s}\in \Sigma _C^{\textrm{rat}}$	rational cluster (Definition 3.9)
$\Sigma _F$	$=\Sigma _{C_F}^{\textrm{rat}}$ , for some extension $F/K$ (Definition 4.6)
$\Sigma _f^z,\Sigma _C^z$	cluster picture centred at $z$ (Definition 3.34)
$\mathfrak{s}\in \Sigma _C^z$	cluster centred at $z$ (Definition 3.33)
$\rho _{\mathfrak{s}}^z,\epsilon _{\mathfrak{s}}^z$	$\rho _{\mathfrak{s}}^z=\min _{r\in \mathfrak{s}}v(r-z)$ , $\epsilon _{\mathfrak{s}}^z=v(c_f)+\sum _{r\in \mathfrak{R}}\rho _{r\wedge \mathfrak{s}}^z$ (Definition 3.35)
$\Sigma ^W$ , $\Sigma ^{nr}$	$\Sigma ^W=\bigcup _{w\in W}\Sigma _C^w$ , $\Sigma ^{nr}\subset \Sigma _{K^{nr}}$ non-removable clusters (Definition 4.20)
$D_{\mathfrak{s}},m_{\mathfrak{s}}$	$D_{\mathfrak{s}}=1$ if $b_{\mathfrak{s}}\epsilon _{\mathfrak{s}}$ odd, $2$ if $b_{\mathfrak{s}}\epsilon _{\mathfrak{s}}$ even; $m_{\mathfrak{s}}=(3-D_{\mathfrak{s}})b_{\mathfrak{s}}$ (Definition 4.6)
$p_{\mathfrak{s}}$	$=1$ if $|\mathfrak{s}|$ is odd, $2$ if $|\mathfrak{s}|$ is even (Definition 4.6)
$\gamma _{\mathfrak{s}}$	$=2$ if $|\mathfrak{s}|$ is even and $\epsilon _{\mathfrak{s}}\!-\!|\mathfrak{s}|\rho _{\mathfrak{s}}$ is odd, $1$ otherwise (Definition 4.6)
$p_{\mathfrak{s}}^0$	$=1$ if $\mathfrak{s}$ is minimal and $\mathfrak{s}\cap K_{\mathfrak{s}}\neq \varnothing$ , $2$ otherwise (Definition 4.6)
$\gamma _{\mathfrak{s}}^0$	$=2$ if $p_{\mathfrak{s}}^0=2$ and $\epsilon _{\mathfrak{s}}$ is odd, 1 otherwise (Definition 4.6)
$s_{\mathfrak{s}}$ , $s_{\mathfrak{s}}^0$	$s_{\mathfrak{s}}=\frac 12(|\mathfrak{s}|\rho _{\mathfrak{s}}+p_{\mathfrak{s}}\rho _{\mathfrak{s}}-\epsilon _{\mathfrak{s}})$ , $s_{\mathfrak{s}}^0=-\epsilon _{\mathfrak{s}}/2+\rho _{\mathfrak{s}}$ (Definition 4.6)
${\overline{g_{\mathfrak{s}}}},{\overline{g_{\mathfrak{s}}^0}},{\overline{f_{\mathfrak{s}}^W}},{\overline{f_{\mathfrak{s}}}}, \tilde{f}_{\mathfrak{s}}$	polynomials in one variable over $k_{\mathfrak{s}}$ (Definitions 4.14 and 4.22)

In Section 5 we explicitly construct proper flat models of hyperelliptic curves and study the conditions for having (minimal) regular models with normal crossings. Here you can find the most used objects and notation.

$\Sigma$	$=\{{\mathfrak{s}}_1,\dots,{\mathfrak{s}}_m\}$ , set of rationally minimal clusters (Section 5.1)
${\mathfrak{s}}_h$	a rationally minimal cluster, element of $\Sigma$ (Section 5.1)
$W$	$=\{w_1,\dots,w_m\}$ , where $w_h$ is a rational centre of ${\mathfrak{s}}_h$ (Section 5.1)
$w_h$	fixed rational centre of ${\mathfrak{s}}_h$ , element of $W$ (Section 5.1)
$w_{hl}$	$=w_h-w_l$ for fixed rational centres $w_h,w_l$ (Section 5.1)
$u_{hl},\rho _{hl}$	$u_{hl}\in O_K^\times$ , $\rho _{hl}\in \mathbb{Z}$ such that $w_{hl}=u_{hl}\pi ^{\rho _{hl}}$ ; $u_{hh}=0$ (Section 5.1)
$M$	matrix associated to a proper rational cluster $\mathfrak{t}\in \Sigma ^W$ (Definition 5.1, Lemma 5.2)
$\stackrel{M}{=}$	change of variable $(x,y,\pi )\stackrel{M}{=}(X,Y,Z)\bullet M^{-1}$ given by $M$ (Section 5.2)
$\delta _M, \sigma _M, X_M$	integer, cone, toric scheme attached to a matrix $M$ (Definitio 5.1)
$m_{\ast \ast }, \tilde{m}_{\ast \ast }$	entries of the matrices $M$ and $M^{-1}$ (Section 5.2)
$X_{\Delta }^h$	$=\bigcup _{\mathfrak{t},M}X_M$ , toric scheme constructed from $\Delta _v^{w_h}$ (Definition 5.1)
$\mathcal{C}_\Delta ^{w}$	proper model of $C^w$ constructed from $\Delta _v^w$ by [1, 3.14]
$\mathcal{C}_\Delta ^{w_h}$	closure of $C\simeq C^{w_h}$ in $X_\Delta ^h$ (Section 5.2)
$R$	$=O_K[X^{\pm 1},Y,Z]/(\pi -X^{\tilde{m}_{13}}Y^{\tilde{m}_{23}}Z^{\tilde{m}_{33}})$ (Section 5.2)
$T_M^{hl}$	$\in R$ , satisfying $x-w_{hl}\stackrel{M}{=}X^{\ast } Y^{\ast } Z^{\ast }T_M^{hl}$ (Section 5.2)
$T_M^h$	$=\prod _{l\neq h}T_M^{hl}\in R$ (Section 5.2)
$\mathcal{F}_M^h$	$\in R$ , equals $Y^{\ast }Z^{\ast }\cdot g_h((X,Y,Z)\bullet M^{-1})$ (Section 5.2)
$V_M^h$	$=\textrm{Spec}\, R[(T_M^h)^{-1}]\subset X_M$ , (Section 5.2)
$U_M^h$	$=\textrm{Spec}\, R[(T_M^h)^{-1}]/(\mathcal{F}_M^h)\subset V_M^h$ , chart of $\mathcal{C}$ (Section 5.2)
$\mathring{X}_{\Delta} ^h,\mathring{\mathcal{C}}_\Delta ^{w_h}$	$\mathring{X}_\Delta ^h=\bigcup _{\mathfrak{t},M} V_M^h\subseteq X_\Delta ^h$ , $\mathring{\mathcal{C}}_\Delta ^{w_h} =\bigcup _{\mathfrak{t},M} U_M^h\subset X_\Delta ^h$ (Section 5.2)
$\mathcal{X}$ , $\mathcal{C}$	$\mathcal{X}=\bigcup _h{X}_\Delta ^h$ , $\mathcal{C}=\bigcup _h\mathring{\mathcal{C}}_\Delta ^{w_h}$ (Section 5.3)
$\hat{\mathfrak{t}}^W$ , $\tilde{\mathfrak{t}}^W$ , $\tilde{\mathfrak{t}}$	sets attached to a rational cluster $\mathfrak{t}$ (Definition 5.15, before Proposition 5.18 and Definition 4.13)
$\bar{X}_{F_{\mathfrak{t}}^{w}}$	$1$ -dimensional closed subscheme of $\mathcal{C}_{\Delta,s}^{w}$ given by $F_{\mathfrak{t}}^{w}$ (Section 5.6)
$\mathring{X}_{F_{\mathfrak{t}}^{w}}$	$=\bar{X}_{F_{\mathfrak{t}}^{w}}\cap\mathring{\mathcal{C}}_{\Delta }^{w}$ (Section 5.6)
$\Gamma _{\mathfrak{t}}$	$\subseteq \mathcal{C}_s$ , glueing of $\mathring{X}_{F_{\mathfrak{t}}^{w}}$ for all $w\in W$ such that $\mathfrak{t}\in \Sigma _C^{w}$ (Section 5.6)

2. Newton polygon

Let $K$ be a complete field with a normalised valuation $v$ , ring of integers $O_K$ , uniformiser $\pi$ , and residue field $ k$ of characteristic $p$ . We fix $\bar{K}$ , an algebraic closure of $K$ , of residue field $\bar{k}$ , and we denote by $K^{\textrm{s}}$ the separable closure of $K$ in $\bar{K}$ . Denote by $K^{nr}$ the maximal unramified extension of $K$ in $K^{\textrm{s}}$ , by $O_{K^{nr}}$ its ring of integers, and by $ k^{\textrm{s}}$ its residue field. Note that $ k^{\textrm{s}}$ is the separable closure of $k$ in $\bar{k}$ . Extend the valuation $v$ to $\bar{K}$ . Finally, write $G_K$ , $G_k$ for the Galois groups $\textrm{Gal}(K^{\textrm{s}}/K)$ , $\textrm{Gal}( k^{\textrm{s}}/ k)$ , respectively.

Notation 2.1. Let $O_{\bar{K}}=\{a\in \bar{K}\mid v(a)\geq 0\}$ . Throughout this paper, given an element $a\in O_{\bar{K}}$ , we will write $a\mod \pi$ for the reduction of $a$ in $\bar{k}$ . Similarly, given a polynomial $h\in O_{\bar{K}}[x_1,\dots,x_n]$ , namely $h=\sum a_{i_1,\dots,i_n}\cdot x_1^{i_1}\cdots x_n^{i_n}$ , we will write $h\mod \pi$ for the polynomial $\sum (a_{i_1,\dots,i_n}\mod \pi )\cdot x_1^{i_1}\cdots x_n^{i_n}\in \bar{k}[x_1,\dots,x_n]$ .

Let $f\in K[x]$ be a non-zero polynomial of degree $d$ , say

\begin{align*} f(x)=\sum _{i=0}^da_ix^i. \end{align*}

The Newton polygon of $f$ , denoted $\texttt{NP}(f)$ , is

\begin{align*} \texttt{NP}(f) = \mbox{lower convex hull}\,\left \{(i,v(a_i))\mid \, i=0,\dots, d, \, a_i\neq 0\right \}\subset{\mathbb{R}}^2. \end{align*}

We recall the following well-known result (see e.g., [17, II.6.3,6.4]).

Theorem 2.2. Let $i_0\lt \ldots \lt i_s=d$ be the set of indices in $\{0,\dots,d\}$ such that the points $(i_0,v(a_{i_0})),\dots,(i_s,v(a_{i_s}))$ are the vertices of $\texttt{NP}(f)$ . For any $j=1,\dots,s$ , denote by $\rho _j$ the slope of the edge of $\texttt{NP}(f)$ which links the points $(i_{j-1},v(a_{i_{j-1}}))$ and $(i_j,v(a_{i_j}))$ . Then $f$ has a unique factorisation over $K$ as a product

\begin{align*} f=a_d \cdot g_0 \cdot g_1\cdots g_s, \end{align*}

where $g_0=x^{i_0}$ and, for all $j=1,\dots, s$ ,

• the polynomials $g_j\in K[x]$ are monic of degree $d_j=i_j-i_{j-1}$ ,

• all the roots of $g_j$ have valuation $-\rho _j$ in $\bar{K}$ .

In particular, $\texttt{NP}(g_j)$ is a segment of slope $\rho _j$ .

Corollary 2.3. With the notation of Theorem 2.2, the polynomial $f$ has exactly $d_j$ roots of valuation $-\rho _j$ for all $j=1,\dots,s$ .

Corollary 2.4. If $f=\sum a_ix^i$ is irreducible of degree $d$ and $a_0\neq 0$ , then $\texttt{NP}(f)$ is a segment linking the points $(0,v(a_0))$ and $(d,v(a_d))$ .

Definition 2.5 (Restriction and reduction). Let $f=\sum _{i=0}^da_ix^i\in K[x]$ and consider an edge $L$ of its Newton polygon $\texttt{NP}(f)$ . Let $(i_1,v(a_{i_1})), (i_2,v(a_{i_2}))$ , $i_1\lt i_2$ be the two endpoints of $L$ . Denote by $\rho$ the slope of $L$ and by $n$ the denominator of $\rho$ . Define the restriction of $f$ to $L$ as

\begin{align*} f|_L\;:\!=\;\sum _{i= 0}^{(i_2-i_1)/n}a_{ni+i_1}x^{i}\in K[x]. \end{align*}

Moreover, we define the reduction of $f$ with respect to $L$ to be the polynomial

\begin{align*}{\overline{f|_L}}\;:\!=\;\pi ^{-c}f|_L(\pi ^{-n\rho } x)\mbox{ mod }\pi \in k[x], \end{align*}

where $c=v(a_{i_1})=v(a_{i_2})+(i_1-i_2)\rho .$

Remark 2.6. These definitions coincide with the ones given in [ 1, Definitions 3.4, 3.5] when the number of variables is $1$ (for suitable choices of basis of the lattices used in the definitions).

Until the end of the section let $f\in K[x]$ , consider a factorisation $f=a_d\cdot g_0\cdot g_1\cdots g_s$ as in Theorem 2.2. Denote by $L_j$ the edge of slope $\rho _j$ of $\texttt{NP}(f)$ , for any $j=1\dots s$ .

Remark 2.7. By the lower convexity of $\texttt{NP}(f)$ , for all $j=1,\dots,s$ , note that ${\overline{f|_{L_j}}}=\bar{c}_j\cdot{\overline{g_j|_{\texttt{NP}(g_j)}}}$ for some $\bar{c}_j\in k^\times$ . In particular they define the same $ k$ -scheme in $\mathbb{G}_{m, k}$ . More precisely, for any $j=1,\dots,s$ , let

\begin{align*} u_j=a_d\cdot \prod _{i=j+1}^{s}g_i(0). \end{align*}

Then $\bar{c}_j=u_j/\pi ^{v(u_j)}\mod \pi$ .

Definition 2.8. We say that $f$ is $\texttt{NP}$ -regular if the $ k$ -scheme

\begin{align*} X_{L_j}\;:\;\{{\overline{f|_{L_j}}}=0\}\subset \mathbb{G}_{m, k} \end{align*}

is smooth for all $j=1,\dots,s$ .

Lemma 2.9. The polynomial $f=a_d\cdot g_0\cdot g_1\cdots g_s$ is $\texttt{NP}$ -regular if and only if $g_j$ is $\texttt{NP}$ -regular for every $j=1,\dots,s$ .

Proof. The Lemma follows from Remark 2.7.

We conclude this section with two examples.

Example 2.10. Let $f=x^{11}+9x^7-3x^6+9x^5+81x-27\in \mathbb{Q}_3[x]$ . Then the Newton polygon of $f$ is

Corollary 2.3 implies that $f$ has $6$ roots of valuation $\frac{1}{3}$ and $5$ roots of valuation $\frac{1}{5}$ . Furthermore, the two polynomials $g_1$ and $g_2$ in the factorisation $f=g_1\cdot g_2$ of Theorem 2.2 turn out to be

\begin{align*} g_1=x^6+9,\qquad g_2=x^5+9x-3. \end{align*}

Finally,

\begin{align*} f|_{L_1}=-3x^2-27=-3\cdot g_1|_{\texttt{NP}(g_1)},\qquad f|_{L_2}=x-3= g_2|_{\texttt{NP}(g_2)}; \end{align*}

and

\begin{align*}{\overline{f|_{L_1}}}=-x^2-1=-(x^2+1)=-{\overline{g_1|_{\texttt{NP}(g_1)}}},\qquad{\overline{f|_{L_2}}}=x-1={\overline{g_2|_{\texttt{NP}(g_2)}}}\qquad \mbox{in }{\mathbb{F}}_3[x]. \end{align*}

Thus $f$ is $\texttt{NP}$ -regular.

Example 2.11. We now show an example of a polynomial that is not $\texttt{NP}$ -regular. Let $f=x^9+12x^6+36x^3+81\in \mathbb{Q}_3[x]$ . Then the Newton polygon of $f$ is

Corollary 2.3 implies that $f$ has $3$ roots of valuation $\frac{2}{3}$ and $6$ roots of valuation $\frac{1}{3}$ . Furthermore, the two polynomials $g_1$ and $g_2$ in the factorisation $f=g_1\cdot g_2$ of Theorem 2.2 are

\begin{align*} g_1=x^3+9,\qquad g_2=x^6+3x^3+9. \end{align*}

Finally,

\begin{align*} f|_{L_1}=36x+81\qquad f|_{L_2}=x^2+12x+36, \end{align*}

\begin{align*} g_1|_{\texttt{NP}(g_1)}=x+9,\qquad g_2|_{\texttt{NP}(g_2)}=x^2+3x+9; \end{align*}

and

\begin{align*}{\overline{f|_{L_1}}}=x+1={\overline{g_1|_{\texttt{NP}(g_1)}}},\qquad{\overline{f|_{L_2}}}=(x+2)^2={\overline{g_2|_{\texttt{NP}(g_2)}}}\qquad \mbox{in }{\mathbb{F}}_3[x]. \end{align*}

Then $f$ is not $\texttt{NP}$ -regular. In fact, in accordance with Lemma 2.9, $g_2$ is not $\texttt{NP}$ -regular.

3. Rational clusters

In this subsection we introduce simple combinatorial objects, that we call rational clusters, attached to a separable polynomial $f\in K[x]$ . Via this new terminology, we will give a characterisation for the $\texttt{NP}$ -regularity, from which the definition of almost rational cluster picture, key condition for the next sections, will follow. In fact, rational clusters are the main objects we will use for the construction of models and the description of integral differentials of hyperelliptic curves in Sections 5 and 6.

From now on, let $f\in K[x]$ be a separable polynomial and denote by $\mathfrak{R}$ the set of its roots in $K^{\textrm{s}}$ and by $c_f$ its leading coefficient. Then

\begin{align*} f(x)=c_f\prod _{r\in \mathfrak{R}}(x-r). \end{align*}

Definition 3.1 ([14, Definition 1.1]). A cluster (for $f$ ) is a non-empty subset $\mathfrak{s}\subseteq \mathfrak{R}$ of the form $\mathcal{D}\cap \mathfrak{R}$ , where $\mathcal{D}$ is a $v$ -adic disc $\mathcal{D}=\{x\in \bar{K}\mid v(x-z)\geq d\}$ for some $z\in \bar{K}$ and $d\in \mathbb{Q}$ . If $|\mathfrak{s}|\gt 1$ we say that $\mathfrak{s}$ is proper and define its depth $d_{\mathfrak{s}}$ to be

\begin{align*} d_{\mathfrak{s}}=\min _{r,r^{\prime}\in \mathfrak{s}}v(r-r'). \end{align*}

Note that every proper cluster is cut out by a disc of the form

\begin{align*} \mathcal{D}=\{x\in \bar{K}\mid v(x-r)\geq d_{\mathfrak{s}}\} \end{align*}

for any $r\in \mathfrak{s}$ .

Definition 3.2 ([14, Definition 1.26]). The cluster picture of $f$ is the collection of its clusters, together with their depths.

We denote by $\Sigma _f$ the set of all clusters of $f$ and by $\mathring{\Sigma }_f$ the subset of $\Sigma _f$ of proper clusters.

Definition 3.3 ([14, Definition 1.3]). If $\mathfrak{s}^{\prime}\subsetneq \mathfrak{s}$ is maximal subcluster, then we say that $\mathfrak{s}^{\prime}$ is a child of $\mathfrak{s}$ and $\mathfrak{s}$ is the parent of $\mathfrak{s}^{\prime}$ , and we write $\mathfrak{s}^{\prime}\lt \mathfrak{s}$ . For any $\mathfrak{s}^{\prime},\mathfrak{s}\in \Sigma _f$ , we write $\mathfrak{s}^{\prime}\leq \mathfrak{s}$ if either $\mathfrak{s}^{\prime}\lt \mathfrak{s}$ or $\mathfrak{s}^{\prime}=\mathfrak{s}$ . Since every cluster $\mathfrak{s}\neq \mathfrak{R}$ has one and only one parent we write $P(\mathfrak{s})$ to refer to the unique parent of $\mathfrak{s}$ .

We say that a proper cluster $\mathfrak{s}$ is minimal if it does not have any proper child.

For two clusters (or roots) ${\mathfrak{s}}_1,{\mathfrak{s}}_2$ , we write ${\mathfrak{s}}_1\wedge{\mathfrak{s}}_2$ for the smallest cluster that contains them.

Definition 3.4 ([14, Definition 1.4]). A cluster $\mathfrak{s}$ is odd/even if its size is odd/even. If $|\mathfrak{s}|=2$ , then we say $\mathfrak{s}$ is a twin. A cluster $\mathfrak{s}$ is übereven if it has only even children.

Definition 3.5 ([14, Definition 1.9]). A centre $z_{\mathfrak{s}}$ of a proper cluster $\mathfrak{s}$ is any element $z_{\mathfrak{s}}\in K^{\textrm{s}}$ such that $\mathfrak{s}=\mathcal{D}\cap \mathfrak{R}$ , where

\begin{align*} \mathcal{D}=\{x\in \bar{K}\mid v(x-z_{\mathfrak{s}})\geq d_{\mathfrak{s}}\}. \end{align*}

Equivalently, $v(r-z_{\mathfrak{s}})\geq d_{\mathfrak{s}}$ for all $r\in \mathfrak{s}$ . The centre of a non-proper cluster $\mathfrak{s}=\{r\}$ is $r$ .

Definition 3.6 ([14, Definition 1.6]). For a proper cluster $\mathfrak{s}$ set

\begin{align*} \nu _{\mathfrak{s}}\;:\!=\;v(c_f)+\sum _{r\in \mathfrak{R}}d_{r\wedge \mathfrak{s}}. \end{align*}

Definition 3.7. We say that $\Sigma _f$ is nested if one of the following equivalent conditions is satisfied:

1. (i) there exists $z\in K^{\textrm{s}}$ such that $z$ is a centre for all proper clusters $\mathfrak{s}\in \Sigma _f$ ;

2. (ii) there is only one minimal cluster in $\Sigma _f$ ;

3. (iii) every non-minimal proper cluster has exactly one proper child.

Definition 3.8. A rational centre of a cluster $\mathfrak{s}$ is any element $w_{\mathfrak{s}}\in K$ such that

\begin{align*} \min _{r\in \mathfrak{s}}v(r-w_{\mathfrak{s}})=\max _{w\in K}\min _{r\in \mathfrak{s}}v(r-w). \end{align*}

If $\mathfrak{s}=\{r\}$ , with $r\in K$ , then $w_{\mathfrak{s}}=r$ .

If $w_{\mathfrak{s}}$ is a rational centre of a proper cluster $\mathfrak{s}$ , we define the radius of $\mathfrak{s}$ to be

\begin{align*} \rho _{\mathfrak{s}}=\min _{r\in \mathfrak{s}}v(r-w_{\mathfrak{s}}). \end{align*}

Definition 3.9. A rational cluster is a cluster cut out by a $v$ -adic disc of the form $\mathcal{D}=\{x\in \bar{K}\mid v(x-w)\geq d\}$ with $w\in K$ and $d\in \mathbb{Q}$ .

The rational cluster picture is the collection of all rational clusters of $f$ together with their radii.

We denote by $\Sigma _f^{\textrm{rat}}\subseteq \Sigma _f$ the set of rational clusters and by $\mathring{\Sigma }_f^{\textrm{rat}}$ the subset of $\Sigma _f^{\textrm{rat}}$ of proper rational clusters.

Lemma 3.10. Let $\mathfrak{s}$ be a proper cluster. Then $d_{\mathfrak{s}}\geq \rho _{\mathfrak{s}}$ .

Proof. First we want to show that

\begin{align*} \min _{r,r^{\prime}\in \mathfrak{s}}v(r-r')=\max _{z\in K^{\textrm{s}}}\min _{r\in \mathfrak{s}}v(r-z). \end{align*}

Clearly $\min _{r,r^{\prime}\in \mathfrak{s}}v(r-r')\leq \max _{z\in K^{\textrm{s}}}\min _{r\in \mathfrak{s}}v(r-z)$ . Let $z_{\mathfrak{s}}\in K^{\textrm{s}}$ such that

\begin{align*} \max _{z\in K^{\textrm{s}}}\min _{r\in \mathfrak{s}}v(r-z)=\min _{r\in \mathfrak{s}}v(r- z_{\mathfrak{s}}). \end{align*}

Then, for any $r,r'\in \mathfrak{s}$ , one has

\begin{align*} v(r-r')\geq \min \{v(r-z_{\mathfrak{s}}), v(r'-z_{\mathfrak{s}})\}\geq \min _{r\in \mathfrak{s}}v(r- z_{\mathfrak{s}}), \end{align*}

and so

\begin{align*} \min _{r,r^{\prime}\in \mathfrak{s}}v(r-r')\geq \max _{z\in K^{\textrm{s}}}\min _{r\in \mathfrak{s}}v(r-z), \end{align*}

as required. From

\begin{align*} d_{\mathfrak{s}}=\min _{r,r^{\prime}\in \mathfrak{s}}v(r-r')=\max _{z\in K^{\textrm{s}}}\min _{r\in \mathfrak{s}}v(r-z)\geq \max _{w\in K}\min _{r\in \mathfrak{s}}v(r-w)=\rho _{\mathfrak{s}}, \end{align*}

the Lemma follows.

Thanks to the previous lemma, the next definition gives, for any cluster $\mathfrak{s}$ , the smallest rational cluster containing it.

Definition 3.11. Given a proper cluster $\mathfrak{s}\in \Sigma _f$ , we define the rationalisation ${\mathfrak{s}}^{\textrm{rat}}$ of $\mathfrak{s}$ to be the smallest rational cluster containing $\mathfrak{s}$ . By definition

\begin{align*}{\mathfrak{s}}^{\textrm{rat}}=\mathfrak{R}\cap \{x\in \bar{K}\mid v(x-w_{\mathfrak{s}})\geq \rho _{\mathfrak{s}}\}, \end{align*}

where $w_{\mathfrak{s}}$ is a rational centre of $\mathfrak{s}$ and $\rho _{\mathfrak{s}}$ is its radius.

The next Lemma will be used in Section 5 to prove the minimality of the regular model with normal crossings we construct.

Lemma 3.12. Let $\mathfrak{s}\in \Sigma _f^{\textrm{rat}}$ be a proper cluster with rational centre $w_{\mathfrak{s}}$ . Let $\mathfrak{s}^{\prime}\in \Sigma _C^{\textrm{rat}}$ be the child of $\mathfrak{s}$ with rational centre $w_{\mathfrak{s}}$ (let $\mathfrak{s}^{\prime}=\varnothing$ if it does not exist). Then $(|\mathfrak{s}|-|\mathfrak{s}^{\prime}|)\rho _{\mathfrak{s}}\in \mathbb{Z}$ .

Proof. As $\mathfrak{s}\in \Sigma _f^{\textrm{rat}}$ , one has $\mathfrak{s}={\mathfrak{s}}^{\textrm{rat}}$ . Let $b_{\mathfrak{s}}$ be the denominator of $\rho _{\mathfrak{s}}$ . Then $b_{\mathfrak{s}}$ divides the degree of the minimal polynomial of $r$ , for any $r\in \mathfrak{s}$ satisfying $v(w_{\mathfrak{s}}-r)=\rho _{\mathfrak{s}}$ . Then $(|\mathfrak{s}|-|\mathfrak{s}^{\prime}|)\rho _{\mathfrak{s}}\in \mathbb{Z}$ , where

\begin{align*} \mathfrak{s}^{\prime}=\mathfrak{R}\cap \{x\in \bar{K}\mid v(x-w_{\mathfrak{s}})\gt \rho _{\mathfrak{s}}\}, \end{align*}

as required.

By definition, a rational cluster is $G_K$ -invariant. Apart from that necessary condition, it is not easy to see whether a proper cluster $\mathfrak{s}$ is also a rational cluster in general. The following remark gives a sufficient condition and shows we have a simple characterisation when $K(\mathfrak{s})/K$ is tamely ramified.

Remark 3.13. If a proper cluster $\mathfrak{s}\in \Sigma _f$ satisfies $d_{\mathfrak{s}}=\rho _{\mathfrak{s}}$ , then a rational centre $w_{\mathfrak{s}}\in K$ of its is also a centre. Hence $\mathfrak{s}$ is a rational cluster and, in particular, is $G_K$ -invariant. On the other hand, if a proper cluster $\mathfrak{s}\in \Sigma _f$ is $G_K$ -invariant and $K(\mathfrak{s})/K$ is tamely ramified, then $\mathfrak{s}$ has a centre $z_{\mathfrak{s}}\in K$ by [14 , Lemma B.1]. Thus $\rho _{\mathfrak{s}}=d_{\mathfrak{s}}$ and $\mathfrak{s}\in \Sigma _f^{\textrm{rat}}$ .

Lemma 3.14. Let $\mathfrak{s}$ be a proper cluster with rational centre $w_{\mathfrak{s}}$ and let $\mathfrak{t}\in \Sigma _f$ satisfying $\mathfrak{t}\supseteq \mathfrak{s}$ . Then $w_{\mathfrak{s}}$ is a rational centre of $\mathfrak{t}$ and $\rho _{\mathfrak{t}}\leq \rho _{\mathfrak{s}}$ . Furthermore, if $\mathfrak{s}$ is a rational cluster and $\mathfrak{t}\supsetneq \mathfrak{s}$ , then $\rho _{\mathfrak{t}}\lt \rho _{\mathfrak{s}}$ .

Proof. It suffices to prove the Lemma for $\mathfrak{t}=P(\mathfrak{s})$ . Hence we first want to show that $\min _{r\in P(\mathfrak{s})}v(r-w_{\mathfrak{s}})=\rho _{P(\mathfrak{s})}$ and $\rho _{P(\mathfrak{s})}\leq \rho _{\mathfrak{s}}$ . Note that

\begin{align*} \min _{r\in P(\mathfrak{s})}v(r-w_{\mathfrak{s}})\leq \max _{w\in K}\min _{r\in P(\mathfrak{s})}v(r-w)=\rho _{P(\mathfrak{s})}. \end{align*}

Moreover,

\begin{align*} \rho _{P(\mathfrak{s})}=\max _{w\in K}\min _{r\in P(\mathfrak{s})}v(r-w)\leq \max _{w\in K}\min _{r\in \mathfrak{s}}v(r-w)=\rho _{\mathfrak{s}}. \end{align*}

If $r\in \mathfrak{s}$ then $v(w_{\mathfrak{s}}-r)\geq \rho _{\mathfrak{s}}$ , by definition of $\rho _{\mathfrak{s}}$ . On the other hand, if $r\in P(\mathfrak{s})\smallsetminus \mathfrak{s}$ then fixing $r'\in \mathfrak{s}$ we have

\begin{align*} v(r-w_{\mathfrak{s}})=v(r-r'+r'-w_{\mathfrak{s}})\geq \min \{v(r-r'),v(r'-w_{\mathfrak{s}})\}\geq \min \{d_{P(\mathfrak{s})},\rho _{\mathfrak{s}}\}\geq \rho _{P(\mathfrak{s})}, \end{align*}

by the previous lemma. Thus $\min _{r\in P(\mathfrak{s})}v(r-w_{\mathfrak{s}})=\rho _{P(\mathfrak{s})}$ , as required.

Now suppose $\mathfrak{s}\in \Sigma _f^{\textrm{rat}}$ with $\mathfrak{t}\supsetneq \mathfrak{s}$ . From Definition 3.8, it follows that

\begin{align*} \{x\in \bar{K}\mid v(x-w_{\mathfrak{s}})\geq \rho _{\mathfrak{s}}\}\cap \mathfrak{R}=\mathfrak{s}\subsetneq \mathfrak{t}\subseteq \{x\in \bar{K}\mid v(x-w_{\mathfrak{s}})\geq \rho _{\mathfrak{t}}\}\cap \mathfrak{R}, \end{align*}

as $w_{\mathfrak{s}}$ is a rational centre of $\mathfrak{t}$ . Thus $\rho _{\mathfrak{t}}\lt \rho _{\mathfrak{s}}$ .

Definition 3.15. We say that a proper rational cluster $\mathfrak{s}\in \Sigma _f^{\textrm{rat}}$ is (rationally) minimal if it does not have any proper rational child.

From Lemma 3.14 it follows that if $W\subseteq K$ such that every minimal rational cluster has a rational centre in $W$ , then all clusters have a rational centre in $W$ . This fact will be key for the construction of the model in Section 5. Another important result is Lemma 3.18, that describes the depth and the radius of $\mathfrak{s}\wedge \mathfrak{s}^{\prime}$ for two rational clusters $\mathfrak{s},\mathfrak{s}^{\prime}$ . To prove it, we need the following two lemmas.

Lemma 3.16. Every cluster $\mathfrak{s}$ with $\rho _{\mathfrak{s}}\lt d_{\mathfrak{s}}$ has no rational subcluster $\mathfrak{s}^{\prime}\subsetneq \mathfrak{s}$ .

Proof. Suppose by contradiction there exists $\mathfrak{s}^{\prime}\in \Sigma _C^{\textrm{rat}}$ , $\mathfrak{s}^{\prime}\subsetneq \mathfrak{s}$ , and fix a rational centre $w_{\mathfrak{s}^{\prime}}$ of $\mathfrak{s}^{\prime}$ . Then $w_{\mathfrak{s}^{\prime}}$ is a rational centre of $\mathfrak{s}$ by the previous lemma. If $|\mathfrak{s}^{\prime}|=1$ , then $w_{\mathfrak{s}^{\prime}}$ is also a centre of $\mathfrak{s}$ and this contradicts $\rho _{\mathfrak{s}}\lt d_{\mathfrak{s}}$ ; so, assume $\mathfrak{s}^{\prime}$ proper. Let $r'\in \mathfrak{s}^{\prime}$ such that $v(r'-w_{\mathfrak{s}^{\prime}})=\rho _{\mathfrak{s}^{\prime}}$ and $r\in \mathfrak{s}$ such that $v(r-w_{\mathfrak{s}^{\prime}})=\rho _{\mathfrak{s}}$ . But then $d_{\mathfrak{s}}\leq v(r-w_{\mathfrak{s}^{\prime}}+w_{\mathfrak{s}^{\prime}}-r')=\rho _{\mathfrak{s}}$ again by Lemma 3.14.

In particular, the Lemma above shows that if $\mathfrak{s}\in \Sigma _f$ and $\mathfrak{s}^{\prime}\in \Sigma _f^{\textrm{rat}}$ is a maximal rational subcluster of $\mathfrak{s}$ , with $\mathfrak{s}^{\prime}\subsetneq \mathfrak{s}$ , then $\mathfrak{s}^{\prime}$ is a child of $\mathfrak{s}$ . Moreover, the parent of a rational cluster is rational.

Lemma 3.17. Let $\mathfrak{s},\mathfrak{s}^{\prime}\in \Sigma _f^{\textrm{rat}}$ such that $\mathfrak{s}^{\prime}\nsubseteq \mathfrak{s}$ . If $w_{\mathfrak{s}}$ is a rational centre of $\mathfrak{s}$ then

\begin{align*} \min _{r\in \mathfrak{s}^{\prime}}v(r-w_{\mathfrak{s}})=\rho _{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}. \end{align*}

Proof. By Lemma 3.14 we have

\begin{align*} \min _{r\in \mathfrak{s}\wedge \mathfrak{s}^{\prime}}v(r-w_{\mathfrak{s}})=\rho _{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}. \end{align*}

Therefore $\min _{r\in \mathfrak{s}^{\prime}}v(w_{\mathfrak{s}}-r)\geq \rho _{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}$ . Suppose by contradiction that

\begin{align*} \min _{r\in \mathfrak{s}^{\prime}}v(r-w_{\mathfrak{s}})\;=\!:\;\rho \gt \rho _{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}. \end{align*}

It follows from Lemma 3.14 that

\begin{align*} \min _{r\in \mathfrak{s}}v(r-w_{\mathfrak{s}})=\rho _{\mathfrak{s}}\gt \rho _{\mathfrak{s}\wedge \mathfrak{s}^{\prime}} \end{align*}

as $\mathfrak{s}^{\prime}\nsubseteq \mathfrak{s}$ . But then there exists $ \tilde{r}\in (\mathfrak{s}\wedge \mathfrak{s}^{\prime})\smallsetminus (\mathfrak{s}\cup \mathfrak{s}^{\prime})$ such that $v(\tilde{r}-w_{\mathfrak{s}})=\rho _{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}$ . Consider the rational cluster

\begin{align*} \mathfrak{t}\;:\!=\;\mathfrak{R}\cap \left \{x\in \bar{K}\mid v(x-w_{\mathfrak{s}})\geq \min \{\rho,\rho _{\mathfrak{s}}\}\right \}\in \Sigma _f^{\textrm{rat}}. \end{align*}

Then $\mathfrak{s},\mathfrak{s}^{\prime}\subseteq \mathfrak{t}$ , but since $\tilde{r}\notin \mathfrak{t}$ we have $\mathfrak{s}\wedge \mathfrak{s}^{\prime}\nsubseteq \mathfrak{t}$ that contradicts the minimality of $\mathfrak{s}\wedge \mathfrak{s}^{\prime}$ .

Lemma 3.18. Let $\mathfrak{t}\in \Sigma _f$ with at least two children in $\Sigma _f^{\textrm{rat}}$ . Then $d_{\mathfrak{t}}=\rho _{\mathfrak{t}}\in \mathbb{Z}$ and $\mathfrak{t}\in \Sigma _f^{\textrm{rat}}$ . More precisely, if $\mathfrak{s},\mathfrak{s}^{\prime}\in \Sigma _f^{\textrm{rat}}$ such that $\mathfrak{s}\subsetneq \mathfrak{s}\wedge \mathfrak{s}^{\prime}\supsetneq \mathfrak{s}^{\prime}$ , then

\begin{align*} \rho _{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}=v(w_{\mathfrak{s}}-w_{\mathfrak{s}^{\prime}})=d_{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}, \end{align*}

where $w_{\mathfrak{s}}$ and $w_{\mathfrak{s}^{\prime}}$ are rational centres of $\mathfrak{s}$ and $\mathfrak{s}^{\prime}$ respectively.

Proof. If $d_{\mathfrak{t}}=\rho _{\mathfrak{t}}$ , then $\mathfrak{t}\in \Sigma _f^{\textrm{rat}}$ by Remark 3.13. Hence it suffices to prove the second statement as $v(w_{\mathfrak{s}}-w_{\mathfrak{s}^{\prime}})\in \mathbb{Z}$ . For our assumptions $\mathfrak{s}^{\prime}\not \subseteq \mathfrak{s}$ . Then by Lemma 3.17 there exists $r\in \mathfrak{s}^{\prime}$ so that $v(r-w_{\mathfrak{s}})=\rho _{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}$ . Thus,

\begin{align*} v(w_{\mathfrak{s}}-w_{\mathfrak{s}^{\prime}})=\min \{v(w_{\mathfrak{s}}-r),v(r-w_{\mathfrak{s}^{\prime}})\}=\rho _{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}, \end{align*}

as $v(r-w_{\mathfrak{s}^{\prime}})\geq \rho _{\mathfrak{s}^{\prime}}\gt \rho _{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}$ by Lemma 3.14. Finally, $d_{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}=\rho _{\mathfrak{s}\wedge \mathfrak{s}^{\prime}}$ follows from Lemma 3.16.

Definition 3.19. For a proper cluster $\mathfrak{s}$ set

\begin{align*} \epsilon _{\mathfrak{s}}\;:\!=\;v(c_f)+\sum _{r\in \mathfrak{R}}\rho _{r\wedge \mathfrak{s}}. \end{align*}

Example 3.20. Let $f=x^{11}-3x^6+9x^5-27\in \mathbb{Q}_3[x]$ . The set of roots of $f$ is

\begin{align*} \mathfrak{R}=\{\sqrt [3]{3}, \zeta _3\sqrt [3]{3},\zeta _3^2\sqrt [3]{3},-\sqrt [3]{3}, -\zeta _3\sqrt [3]{3},-\zeta _3^2\sqrt [3]{3},\sqrt [5]{3}, \zeta _5\sqrt [5]{3}, \zeta _5^2\sqrt [5]{3}, \zeta _5^3\sqrt [5]{3}, \zeta _5^4\sqrt [5]{3}\}, \end{align*}

where $\zeta _q$ is a primitive $q$ th root of unity for $q=3,5$ . Then the proper clusters of $f$ are

\begin{align*}{\mathfrak{s}}_1=\{\sqrt [3]{3}, \zeta _3\sqrt [3]{3},\zeta _3^2\sqrt [3]{3}\},\quad{\mathfrak{s}}_2=\{-\sqrt [3]{3}, -\zeta _3\sqrt [3]{3},-\zeta _3^2\sqrt [3]{3}\},\quad{\mathfrak{s}}_3={\mathfrak{s}}_1\cup{\mathfrak{s}}_2,\quad \mathfrak{R} \end{align*}

with $d_{{\mathfrak{s}}_1}=d_{{\mathfrak{s}}_2}=\frac{5}{6}$ , $d_{{\mathfrak{s}}_3}=\frac{1}{3}$ and $d_{\mathfrak{R}}=\frac{1}{5}$ . The graphic representation of the cluster picture of $f$ is then

where the subscripts of clusters (represented as circles) are their depths.

Furthermore, note that $0$ is a rational centre for all proper clusters and we have $\rho _{{\mathfrak{s}}_1}=\rho _{{\mathfrak{s}}_2}=\rho _{{\mathfrak{s}}_3}=\frac{1}{3}$ and $\rho _{\mathfrak{R}}=\frac{1}{5}$ .

Finally, for every cluster $\mathfrak{s}$ we can also compute $\nu _{\mathfrak{s}}$ and $\epsilon _{\mathfrak{s}}$ , that are

\begin{align*} \nu _{{\mathfrak{s}}_1}=\nu _{{\mathfrak{s}}_2}=\frac{9}{2},\quad \nu _{{\mathfrak{s}}_3}=\epsilon _{{\mathfrak{s}}_1}=\epsilon _{{\mathfrak{s}}_2}=\epsilon _{{\mathfrak{s}}_3}=3,\quad \nu _{\mathfrak{R}}=\epsilon _{\mathfrak{R}}=\frac{11}{5}. \end{align*}

Example 3.21. Let $f=x^9+12x^6+36x^3+81\in \mathbb{Q}_3[x]$ and fix an isomorphism ${\overline{\mathbb{Q}}}_3\simeq{\mathbb{C}}$ . Then the set of roots of $f$ is

\begin{align*} \mathfrak{R}=\{\sqrt [3]{3^2}, \zeta _3\sqrt [3]{3^2},\zeta _3^2\sqrt [3]{3^2},\zeta _9\sqrt [3]{3}, \zeta _9^2\sqrt [3]{3},\zeta _9^4\sqrt [3]{3},\zeta _9^5\sqrt [3]{3},\zeta _9^7\sqrt [3]{3},\zeta _9^8\sqrt [3]{3}\}, \end{align*}

where $\zeta _q=e^{2\pi i/q}$ is a primitive $q$ th root of unity for $q=3,9$ . Then the proper clusters of $f$ are

\begin{align*} \begin{array}{c}{\mathfrak{s}}_1=\{\sqrt [3]{3^2}, \zeta _3\sqrt [3]{3^2},\zeta _3^2\sqrt [3]{3^2}\},\quad{\mathfrak{s}}_2=\{\zeta _9\sqrt [3]{3}, \zeta _9^4\sqrt [3]{3}, \zeta _9^7\sqrt [3]{3}\},\\ {\mathfrak{s}}_3=\{\zeta _9^2\sqrt [3]{3}, \zeta _9^5\sqrt [3]{3}, \zeta _9^8\sqrt [3]{3}\},\quad{\mathfrak{s}}_4={\mathfrak{s}}_2\cup{\mathfrak{s}}_3,\quad \mathfrak{R} \end{array} \end{align*}

with $d_{{\mathfrak{s}}_1}=\frac{7}{6}$ , $d_{{\mathfrak{s}}_2}=d_{{\mathfrak{s}}_3}=\frac{5}{6}$ , $d_{{\mathfrak{s}}_4}=\frac{1}{2}$ , and $d_{\mathfrak{R}}=\frac{1}{3}$ . The cluster picture of $f$ is then

It is easy to see that $0$ is a rational centre for all proper clusters and that $\rho _{{\mathfrak{s}}_1}=\frac{2}{3}$ , $\rho _{{\mathfrak{s}}_2}=\rho _{{\mathfrak{s}}_3}=\rho _{{\mathfrak{s}}_4}=\rho _{\mathfrak{R}}=\frac{1}{3}$ . Finally,

\begin{align*} \nu _{{\mathfrak{s}}_1}=\frac{11}{2},\quad \nu _{{\mathfrak{s}}_2}=\nu _{{\mathfrak{s}}_3}=5,\quad \nu _{{\mathfrak{s}}_4}=4,\quad \nu _{\mathfrak{R}}=3;\qquad \epsilon _{{\mathfrak{s}}_1}=4,\quad \epsilon _{{\mathfrak{s}}_2}=\epsilon _{{\mathfrak{s}}_3}=\epsilon _{{\mathfrak{s}}_4}=\epsilon _{\mathfrak{R}}=3. \end{align*}

The goal of this section is to describe the $\texttt{NP}$ -regularity of $f\in K[x]$ (and its translations) in terms of conditions on its cluster picture.

Notation 3.22. If $p\gt 0$ , we denote by $|\cdot |_p$ the standard $p$ -adic absolute value attached to $\mathbb{Q}$ , that is $|a|_p=p^{-v_p(a)}$ for all $a\in \mathbb{Q}$ . If $p=0$ , then we write $|\cdot |_p$ for the function on $\mathbb{Q}$ identically equal to $1$ , that is $|a|_p=1$ for all $a\in \mathbb{Q}$ .

Lemma 3.23. Suppose that $x\nmid f$ and that $\texttt{NP}(f)$ is a segment $L$ of slope $-\rho$ . Let $n$ be the denominator of $\rho$ . Then $f$ is $\texttt{NP}$ -regular if and only if all proper clusters $\mathfrak{s}\in \Sigma _f$ with $|\mathfrak{s}|\gt |\rho |_p$ satisfy $d_{\mathfrak{s}}=\rho$ .

More precisely:

1. (i) If $\mathfrak{s}\in \Sigma _f$ with $|\mathfrak{s}|\gt |\rho |_p$ but $d_{\mathfrak{s}}\gt \rho$ , then $\overline{f|_L}$ has a non-zero multiple root $\bar{u}=\frac{r^n}{\pi ^{n\rho }} \mod \pi$ , for some (any) $r\in \mathfrak{s}$ .

2. (ii) The multiplicity of a root $\bar{u}\in \bar{k}^\times$ of $\overline{f|_L}$ equals $|{\mathfrak{s}}^0|/n$ , where

   \begin{align*}{\mathfrak{s}}^0=\left \{r\in \mathfrak{R}\mid \bar{u}=\tfrac{r^n}{\pi ^{n\rho }}\mod \pi \right \}. \end{align*}

3. (iii) All multiple roots of $\overline{f|_L}$ come from clusters $\mathfrak{s}$ as described in (i).

Proof. Let $q$ be the highest power of $p$ dividing $n$ (set $q=1$ if $p=0$ ). Let $m=n/q$ so that $p\nmid m$ . Let $\mathfrak{R}=\{r_i\mid i=1,\dots,D\}$ be the (multi-)set of roots of $f$ , where $D\;:\!=\;\deg f$ . Fix some choice of $\sqrt [n]{\pi }$ and define $\bar{u}_i\in \bar{k}^\times$ as $\bar{u}_i=r_i/\pi ^\rho \mod \pi$ , for all $i=1,\dots, D$ . Firstly, note that there exists a proper cluster $\mathfrak{s}$ with $|\mathfrak{s}|\gt |\rho |_p$ and $d_{\mathfrak{s}}\gt \rho$ if and only if there exists a subset $I\subseteq \{1,\dots,D\}$ of size $|I|\gt q$ such that $\bar{u}_{i_1}=\bar{u}_{i_2}$ for all $i_1,i_2\in I$ . Indeed, given $\mathfrak{s}$ , then $I=\{i\in \{1,\dots,D\}\mid r_i\in \mathfrak{s}\}$ , while given $I$ , then $\mathfrak{s}=\{r_i\mid \bar{u}_i=\bar{u}_{i_0},\text{ for any }i_0\in I\}$ . Secondly, recall that $f$ is not $\texttt{NP}$ -regular if and only if $\overline{f|_L}$ has a multiple root in $\bar{k}^\times$ . Therefore we will prove that $\overline{f|_L}$ has a non-zero multiple root if and only if there exists a subset $I\subseteq \{1,\dots,D\}$ with size $|I|\gt q$ and such that $\bar{u}_{i_1}=\bar{u}_{i_2}$ for all $i_1,i_2\in I$ .

Note that for the lower convexity of $\texttt{NP}(f)=L$ , we have

\begin{align*}{\overline{f|_L}}(x^n)=\pi ^{-(v(c_f)+D\rho )}f(\pi ^\rho x)\mod \pi. \end{align*}

Hence $\{\bar{u}_i \mid i=1,\dots,D\}$ is the multiset of roots of ${\overline{f|_L}}(x^n)$ . Then there exists an $n$ -to- $1$ map

\begin{align*} \bar \phi &: \{\bar{u}_i\}\longrightarrow{\{\bar{w}_j\}},\\ &\quad \bar{u}_i\mapsto \bar{u}_i^m., \end{align*}

where $\{\bar{w}_j\mid j=1,\dots,D/n\}$ is the multiset of roots of $\overline{f|_L}$ . Note that $\bar{w}_j\neq 0$ for all $j=1,\dots,D/n$ , so all roots of $\overline{f|_L}$ are non-zero.

Now, suppose that $f$ is not $\texttt{NP}$ -regular. We want to show that there exists a subset $I\subset \{1,\dots,D\}$ with $|I|\gt q$ such that $\bar{u}_{i_1}=\bar{u}_{i_2}$ for all $i_1,i_2\in I$ . Since $f$ is not $\texttt{NP}$ -regular, its reduction $\overline{f|_L}$ has a (non-zero) multiple root. Then there exist $j_1,j_2\in \{1,\dots,D/n\}$ so that $\bar{w}_{j_1}=\bar{w}_{j_2}\;=\!:\;\bar{w}$ . Hence, by the definition of $\bar \phi$ , for some (any) $\bar{u}\in \bar \phi ^{-1}(\bar{w})$ , there are at least $2 q$ $\bar{u}_i$ ’s with $\bar{u}_i=\bar{u}$ . Let $I$ denote the set of their indices. Then $|I|\geq 2q\gt q$ and $\bar{u}_{i_1}=\bar{u}_{i_2}$ for all $i_1,i_2\in I$ , as required.

On the other hand, suppose that there exists a subset $I\subset \{1,\dots,D\}$ with $|I|\gt q$ and such that $\bar{u}_{i_1}=\bar{u}_{i_2}$ for all $i_1,i_2\in I$ . We want to show that $\overline{f|_L}$ has a multiple root, that is there exist two indices $j_1,j_2\in \{1,\dots,D/n\}$ such that $\bar{w}_{j_1}=\bar{w}_{j_2}$ . Suppose not and let $j\in \{1,\dots,D/n\}$ such that $\bar{w}_j=\bar{u}_i^m=\bar \phi (\bar{u}_i)$ for some (all) $i\in I$ . Then the polynomial $x^n-\bar{w}_j=(x^m-\bar{w}_j)^{q}\in \bar{k}[x]$ , factor of ${\overline{f|_L}}(x^n)$ , should have a root of order $|I|\gt q$ . This would imply $x^m-\bar{w}_j$ is inseparable, a contradiction as $p\nmid m$ .

The parts (i), (ii) and (iii) of the Lemma follow from above:

1. (i) Given a proper cluster $\mathfrak{s}\in \Sigma _f$ with $|\mathfrak{s}|\gt |\rho |_p$ and $d_{\mathfrak{s}}\gt \rho$ , we showed that $\overline{f|_L}$ has a non-zero multiple root $\bar{w}_j=\bar{u}_i^n={r_i^n}/{\pi ^{n\rho }}\mod \pi$ , where $r_i$ is any root in $\mathfrak{s}$ .

2. (ii) By the definition of $\bar{\phi }$ , given $\bar{w}\in \bar{k}$ , the number of $\bar{w}_j$ ’s such that $\bar{w}_j=\bar{w}$ equals $|{\mathfrak{s}}^0|/n$ , where ${\mathfrak{s}}^0=\{r_i\mid \bar{u}_i^n=\bar{w}\}$ .

3. (iii) Given a (non-zero) multiple root $\bar{w}$ of $\overline{f|_L}$ we showed that there exists $I\subseteq \{1,\dots,D\}$ , with $|I|\gt q$ and $\bar{u}_{i_1}=\bar{u}_{i_2}$ for any $i_1,i_2\in I$ , such that $\bar{u}_i^n=\bar{w}$ for all $i\in I$ . The set $\mathfrak{s}=\{r_i\mid \bar{u}_i=\bar{u}_{i_0},\text{ for any }i_0\in I\}$ is a proper cluster as in (i).

Theorem 3.24. Let $w\in K$ and $f_w(x)=f(x+w)$ . For all clusters $\mathfrak{s}\in \Sigma _f$ define $\lambda _{\mathfrak{s}}=\min _{r\in \mathfrak{s}}v(r-w)$ , and let $b$ be the denominator of $\lambda _{\mathfrak{s}}$ . Then $f_w$ is $\texttt{NP}$ -regular if and only if all proper clusters $\mathfrak{s}\in \Sigma _f$ with $|\mathfrak{s}|\gt |\lambda _{\mathfrak{s}}|_p$ have $d_{\mathfrak{s}}=\lambda _{\mathfrak{s}}$ .

More precisely:

1. (i) Let $\mathfrak{s}\in \Sigma _f$ with $|\mathfrak{s}|\gt |\lambda _{\mathfrak{s}}|_p$ but $d_{\mathfrak{s}}\gt \lambda _{\mathfrak{s}}$ , and let $r\in \mathfrak{s}$ with $v(r-w)=\lambda _{\mathfrak{s}}$ . Then $\overline{f_w|_L}$ has a non-zero multiple root $\bar{u}=\frac{(r-w)^b}{\pi ^{b\lambda _{\mathfrak{s}}}}\mod \pi$ , where $L$ is the edge of $\texttt{NP}(f_w)$ of slope $-\lambda _{\mathfrak{s}}$ .

2. (ii) Let $L$ be an edge of $\texttt{NP}(f_w)$ of slope $-\lambda$ . Let $l$ be the denominator of $\lambda$ . The multiplicity of a root $\bar{u}\in \bar{k}^\times$ of $\overline{f_w|_L}$ equals $|{\mathfrak{s}}^0|/l$ , where

   \begin{align*}{\mathfrak{s}}^0=\big \{r\in \mathfrak{R}\mid v(r-w)=\lambda \quad \text{and}\quad \bar{u}=\tfrac{(r-w)^l}{\pi ^{l\lambda }}\mod \pi \big \}. \end{align*}

3. (iii) For every edge $L$ of $\texttt{NP}(f_w)$ , the multiple roots of $\overline{f_w|_L}$ come from proper clusters $\mathfrak{s}$ for $f$ as described in (i).

Proof. Let $\mathfrak{R}_w$ be the set of roots of $f_w$ . Note that we have a natural bijection $\mathfrak{R}\rightarrow \mathfrak{R}_w$ , $r\mapsto r-w$ , which induces a bijective function $\psi :\Sigma _{f}\rightarrow \Sigma _{f_w}$ , sending

\begin{align*} \mathfrak{s}=\mathfrak{R}\cap \{x\in \bar{K}\mid v(x-z)\gt d\}\quad \mapsto \quad \psi (\mathfrak{s})=\mathfrak{R}_w\cap \{x\in \bar{K}\mid v(x+w-z)\gt d\}. \end{align*}

In particular, if $\mathfrak{s}\in \Sigma _{f}$ , $|\mathfrak{s}|=|\psi (\mathfrak{s})|$ , $d_{\mathfrak{s}}=d_{\psi (\mathfrak{s})}$ and

\begin{align*} \lambda _{\mathfrak{s}}=\min _{r\in \mathfrak{s}}v(r-w)=\min _{r\in \psi (\mathfrak{s})}v(r). \end{align*}

Hence it suffices to show the theorem for $w=0$ .

Assume $w=0$ . Let $f=c_f\cdot g_0\cdot g_1\dots g_t$ be a factorisation of Theorem 2.2. Note that if $t=0$ , then either $f\in K$ or $f\in Kx$ . In both cases, $f$ is clearly $\texttt{NP}$ -regular and has no proper clusters. Then assume $t\gt 0$ and let $-\rho _i$ be the slope of $\texttt{NP}(g_i)$ for any $i=1,\dots,t$ . Denote by $\mathfrak{R}$ the set of roots of $f$ and by $\mathfrak{R}_i$ the set of roots of $g_i$ for $i=0,\dots,t$ . Note that the $\mathfrak{R}_i$ ’s are pairwise disjoint. From Remark 2.7, for every edge $L$ of $\texttt{NP}(f)$ there exists $i$ such that ${\overline{f|_L}}=\bar{c}_i\cdot{\overline{g_i|_{\texttt{NP}(g_i)}}}$ for some $\bar{c}_i\in k^\times$ . Hence, by Lemma 2.9 and Lemma 3.23, we need to prove that there exists a proper cluster $\mathfrak{s}\in \Sigma _f$ such that $|\mathfrak{s}|\gt |\lambda _{\mathfrak{s}}|_p$ and $d_{\mathfrak{s}}\gt \lambda _{\mathfrak{s}}$ if and only if for some $i=1,\dots,t$ there exists a proper cluster ${\mathfrak{s}}_i\in \Sigma _{g_i}$ such that $|{\mathfrak{s}}_i|\gt |\lambda _{{\mathfrak{s}}_i}|_p=|\rho _i|_p$ and $d_{{\mathfrak{s}}_i}\gt \lambda _{{\mathfrak{s}}_i}=\rho _i$ . We will show that one can choose $\mathfrak{s}={\mathfrak{s}}_i$ .

First, note that if $\mathfrak{s}$ is a proper cluster, then $\mathfrak{s}\not \subseteq \mathfrak{R}_0$ , as $|\mathfrak{R}_0|\leq 1$ . Furthermore, if $\mathfrak{s}\in \Sigma _f$ contains roots of different valuations, that is $\mathfrak{s}\nsubseteq \mathfrak{R}_i$ for all $i$ , then

\begin{align*} d_{\mathfrak{s}}=\min _{r,r^{\prime}\in \mathfrak{s}}v(r-r')=\min _{r\in \mathfrak{s}}v(r)=\lambda _{\mathfrak{s}}=\min \{\rho _i\mid \mathfrak{R}_i\cap \mathfrak{s}\neq \varnothing \}. \end{align*}

Now suppose there exists a proper cluster $\mathfrak{s}\in \Sigma _f$ such that $|\mathfrak{s}|\gt |\lambda _{\mathfrak{s}}|_p$ and $d_{\mathfrak{s}}\gt \lambda _{\mathfrak{s}}$ . For the observation above, the inequality $d_{\mathfrak{s}}\gt \lambda _{\mathfrak{s}}$ implies that $\mathfrak{s}\subseteq \mathfrak{R}_i$ for some $i=1,\dots,t$ . Let $\mathcal{D}$ be the $v$ -adic disc such that $\mathfrak{s}=\mathcal{D}\cap \mathfrak{R}$ . Since $\mathfrak{s}\subseteq \mathfrak{R}_i$ , one has $\mathfrak{s}=\mathcal{D}\cap \mathfrak{R}_i$ which means that $\mathfrak{s}\in \Sigma _{g_i}$ , as required.

Finally suppose that for some $i=1,\dots,s$ , there exists a proper cluster ${\mathfrak{s}}_i\in \Sigma _{g_i}$ such that $|{\mathfrak{s}}_i|\gt |\rho _i|_p$ and $d_{{\mathfrak{s}}_i}\gt \rho _i$ . Let $r_i\in{\mathfrak{s}}_i$ . Then

\begin{align*}{\mathfrak{s}}_i=\{x\in \bar{K} \mid v(x-r_i)\geq d_{{\mathfrak{s}}_i}\}\cap \mathfrak{R}_i. \end{align*}

Consider the cluster $\mathfrak{s}\;:\!=\;\{x\in \bar{K} \mid v(x-r_i)\geq d_{{\mathfrak{s}}_i}\}\cap \mathfrak{R}$ of $f$ . Clearly ${\mathfrak{s}}_i\subseteq \mathfrak{s}$ . Therefore

\begin{align*} \lambda _{{\mathfrak{s}}_i}=\min _{r\in{\mathfrak{s}}_i}v(r)\geq \min _{r\in \mathfrak{s}}v(r)=\lambda _{\mathfrak{s}}, \end{align*}

which implies

\begin{align*} d_{\mathfrak{s}}=d_{{\mathfrak{s}}_i}\gt \rho _i=\lambda _{{\mathfrak{s}}_i}\geq \lambda _{\mathfrak{s}}, \end{align*}

where $d_{\mathfrak{s}}=d_{{\mathfrak{s}}_i}$ by construction. Again, from the observation above, the inequality $d_{\mathfrak{s}}\gt \lambda _{\mathfrak{s}}$ implies that $\mathfrak{s}$ is contained in $\mathfrak{R}_j$ for some $j$ . As $\mathfrak{s}\cap \mathfrak{R}_i\supseteq{\mathfrak{s}}_i\cap \mathfrak{R}_i={\mathfrak{s}}_i$ , we must have $\mathfrak{s}\subseteq \mathfrak{R}_i$ . Thus $\mathfrak{s}={\mathfrak{s}}_i$ , that concludes the proof.

Corollary 3.25. Let $f\in K[x]$ be a separable polynomial. Let $w\in K$ and $f_w(x)=f(x+w)$ . Then $f_w$ is $\texttt{NP}$ -regular if and only if all proper clusters $\mathfrak{s}\in \Sigma _f$ have rational centre $w$ and those with $|\mathfrak{s}|\gt |\rho _{\mathfrak{s}}|_p$ satisfy $d_{\mathfrak{s}}=\rho _{\mathfrak{s}}$ .

Proof. If $f_w$ is $\texttt{NP}$ -regular, then, from the previous theorem, all proper clusters $\mathfrak{s}\in \Sigma _f$ with $|\mathfrak{s}|\gt |\lambda _{\mathfrak{s}}|_p$ have $d_{\mathfrak{s}}=\lambda _{\mathfrak{s}}$ , where $\lambda _{\mathfrak{s}}=\min _{r\in \mathfrak{s}}v(r-w)$ . First let $\mathfrak{s}\in \Sigma _f$ proper and assume $|\mathfrak{s}|\gt |\lambda _{\mathfrak{s}}|_p$ . Then

\begin{align*} d_{\mathfrak{s}}=\lambda _{\mathfrak{s}}=\min _{r\in \mathfrak{s}}v(r-w)\leq \max _{z\in K}\min _{r\in \mathfrak{s}}v(r-z)=\rho _{\mathfrak{s}}\leq d_{\mathfrak{s}}, \end{align*}

so $d_{\mathfrak{s}}=\lambda _{\mathfrak{s}}=\rho _{\mathfrak{s}}$ , and $w$ is a rational centre of $\mathfrak{s}$ . Now assume $|\mathfrak{s}|\leq |\lambda _{\mathfrak{s}}|_p$ . In particular, $p\gt 0$ and $\lambda _{\mathfrak{s}}\notin \mathbb{Z}$ , and so

\begin{align*} \min _{r\in \mathfrak{s}}v(r-w)=\lambda _{\mathfrak{s}}\neq v(w-w_{\mathfrak{s}}), \end{align*}

where $w_{\mathfrak{s}}$ is a rational centre of $\mathfrak{s}$ . Let $r\in \mathfrak{s}$ such that $v(r-w)=\lambda _{\mathfrak{s}}$ . Then

\begin{align*} \rho _{\mathfrak{s}}\leq v(r-w+w-w_{\mathfrak{s}})=\min \{\lambda _{\mathfrak{s}},v(w-w_{\mathfrak{s}})\}\leq \lambda _{\mathfrak{s}}. \end{align*}

Clearly

\begin{align*} \rho _{\mathfrak{s}}=\max _{z\in K}\min _{r\in \mathfrak{s}}v(r-z)\geq \min _{r\in \mathfrak{s}}v(r-w)=\lambda _{\mathfrak{s}}, \end{align*}

that implies $\rho _{\mathfrak{s}}=\lambda _{\mathfrak{s}}=\min _{r\in \mathfrak{s}}v(r-w)$ . Hence $w$ is a rational centre of $\mathfrak{s}$ .

On the other hand, suppose that all proper clusters $\mathfrak{s}\in \Sigma _f$ have rational centre $w\in K$ and those with $|\mathfrak{s}|\gt |\rho _{\mathfrak{s}}|_p$ satisfy $d_{\mathfrak{s}}=\rho _{\mathfrak{s}}$ . Then $\rho _{\mathfrak{s}}=\min _{r\in \mathfrak{s}}v(r-w)$ for any $\mathfrak{s}\in \Sigma _f$ . Thus $f_w$ is $\texttt{NP}$ -regular again by Theorem 3.24.

The next definition, which is the main (and only, if $p\neq 2$ ) condition for our explicit construction of the minimal regular model of a hyperelliptic curve given by $y^2=f(x)$ , follows from the statement of Corollary 3.25.

Definition 3.26. We say that $f$ has an almost rational cluster picture if all proper clusters $\mathfrak{s}\in \Sigma _f$ with $|\mathfrak{s}|\gt |\rho _{\mathfrak{s}}|_p$ have $d_{\mathfrak{s}}=\rho _{\mathfrak{s}}$ .

Corollary 3.25 shows that $f$ has a translation which is $\texttt{NP}$ -regular if and only if $f$ has an almost rational cluster picture and there exists $w\in K$ that is a rational centre of all clusters.

In the following we give different characterisations of the previous definition.

Corollary 3.27. Suppose that $K(\mathfrak{R})/K$ is a tamely ramified extension. Then $f$ has an almost rational cluster picture if and only if every proper cluster $\mathfrak{s}\in \Sigma _f$ is $G_K$ -invariant.

Proof. Since $K(\mathfrak{R})/K$ is tamely ramified, every cluster $\mathfrak{s}\in \Sigma _f$ has $|\rho _{\mathfrak{s}}|_p\leq 1$ . Therefore, the Corollary follows from Remark 3.13.

Corollary 3.28. Suppose that $K(\mathfrak{R})/K$ is a tamely ramified extension. Then $f_w$ is $\texttt{NP}$ -regular for some $w\in K$ if and only if $\Sigma _f$ is nested.

Proof. First note that every cluster $\mathfrak{s}\in \Sigma _f$ has $|\rho _{\mathfrak{s}}|_p\leq 1$ , as $K(\mathfrak{R})/K$ is tamely ramified. Therefore, from Corollary 3.25, we need to prove that $\Sigma _f$ is nested if and only if all clusters $\mathfrak{s}\in \Sigma _f$ have $d_{\mathfrak{s}}=\rho _{\mathfrak{s}}$ and rational centre $w$ , for some $w\in K$ . But this follows from Remark 3.13.

Corollary 3.29. The polynomial $f$ has an almost rational cluster picture if and only if for every $r\in \mathfrak{R}\setminus K$ , there exists $w\in K$ so that $r_w^b\;:\!=\;\frac{(r-w)^b}{\pi ^{b\cdot v(r-w)}}\mod \pi$ is a simple root of $f_w|_L$ , where $b$ is the denominator of $v(r-w)$ , $f_w(x)=f(x+w)$ and $L$ is the edge of $\texttt{NP}(f_w)$ of slope $-v(r-w)$ .

Proof. Fix $\tilde{r}\in \mathfrak{R}\setminus K$ and let $\mathfrak{s}$ be the smallest proper cluster containing $\tilde{r}$ . Let $w_{\mathfrak{s}}$ be a rational centre of $\mathfrak{s}$ . Note that $v(\tilde{r}-w_{\mathfrak{s}})=\rho _{\mathfrak{s}}=\min _{r\in \mathfrak{s}} v(r-w_{\mathfrak{s}})$ , for the choice of $\mathfrak{s}$ , as $\tilde{r}\notin K$ . Moreover, for any proper cluster $\mathfrak{t}$ containing $\tilde{r}$ , we have $\mathfrak{s}\subseteq \mathfrak{t}$ . In particular, $w_{\mathfrak{s}}$ is a rational centre of all such clusters. Let $L$ be the edge of $\texttt{NP}(f_{w_{\mathfrak{s}}})$ of slope $-\rho _{\mathfrak{s}}$ . Theorem 3.24 shows that $\tilde{r}_{w_{\mathfrak{s}}}^{b_{\mathfrak{s}}}$ is a multiple root of $f_{w_{\mathfrak{s}}}|_L$ if and only if there exists $\mathfrak{t}\in \Sigma _f$ such that $\tilde{r}\in \mathfrak{t}$ , $|\mathfrak{t}|\gt |\rho _{\mathfrak{t}}|_p$ and $d_{\mathfrak{t}}\gt \rho _{\mathfrak{t}}$ . Therefore, if $f$ has an almost rational cluster picture, then $\tilde{r}_{w_{\mathfrak{s}}}^{b_{\mathfrak{s}}}$ is a simple root.

Suppose there exists $\mathfrak{t}\in \Sigma _f$ such that $|\mathfrak{t}|\gt |\rho _{\mathfrak{t}}|_p$ and $d_{\mathfrak{t}}\gt \rho _{\mathfrak{t}}$ . Then $\mathfrak{t}\cap K=\varnothing$ . By Theorem 3.24, it remains to show that for any $w\in K$ , we have $|\mathfrak{t}|\gt |\lambda _{\mathfrak{t}}|_p$ and $d_{\mathfrak{t}}\gt \lambda _{\mathfrak{t}}$ , where $\lambda _{\mathfrak{t}}=\min _{r\in \mathfrak{t}}v(r-w)$ . First note $d_{\mathfrak{t}}\gt \rho _{\mathfrak{t}}\geq \lambda _{\mathfrak{t}}$ . Moreover, in the proof of Corollary 3.25, we saw that if $|\mathfrak{t}|\leq |\lambda _{\mathfrak{t}}|_p$ then $\rho _{\mathfrak{t}}=\lambda _{\mathfrak{t}}$ and so $|\mathfrak{t}|\leq |\rho _{\mathfrak{t}}|_p$ ; but $|\mathfrak{t}|\gt |\rho _{\mathfrak{t}}|_p$ , thus $|\mathfrak{t}|\gt |\lambda _{\mathfrak{t}}|_p$ .

Lemma 3.30. Suppose $f$ has an almost rational cluster picture. Let $\mathfrak{s}\in \Sigma _f$ proper. If $d_{\mathfrak{s}}\gt \rho _{\mathfrak{s}}$ , then $p\gt 0$ and $|\mathfrak{s}|$ is a $p$ -power. In particular, if $w_{\mathfrak{s}}$ is a rational centre of $\mathfrak{s}$ , for any $r\in \mathfrak{s}$ , the elements $r-w_{\mathfrak{s}}$ are all the roots of a monic polynomial with coefficients in $K^{\textrm{s}}$ , and constant term $c$ such that $|v(c)|_p\geq 1$ .

Proof. Let $\mathfrak{s}\in \Sigma _f$ proper, with $d_{\mathfrak{s}}\gt \rho _{\mathfrak{s}}$ . Since $f$ has an almost rational cluster picture, we must have $|\mathfrak{s}|\leq |\rho _{\mathfrak{s}}|_p$ . Since $\mathfrak{s}$ is proper, $p\gt 0$ . Let $b_{\mathfrak{s}}$ be the denominator of $\rho _{\mathfrak{s}}$ . Then $v_p(b_{\mathfrak{s}})\gt 1$ . Fix a rational centre $w_{\mathfrak{s}}$ of $\mathfrak{s}$ and a root $r\in \mathfrak{s}$ such that $v(r-w_{\mathfrak{s}})=\rho _{\mathfrak{s}}$ . Consider $\mathfrak{s}^{\prime}=\{x\in \mathfrak{R}\mid v(x-r)\gt \rho _{\mathfrak{s}}\}$ . Then $\mathfrak{s}\subseteq \mathfrak{s}^{\prime}\leq{\mathfrak{s}}^{\textrm{rat}}$ and $|\mathfrak{s}^{\prime}|\leq |\rho _{\mathfrak{s}}|_p$ (as $d_{\mathfrak{s}^{\prime}}\gt \rho _{\mathfrak{s}}=\rho _{\mathfrak{s}^{\prime}}$ ). Let $L$ be the Galois closure of $K(r)$ . Let $H$ be the wild inertia subgroup of $\textrm{Gal}(L/K)$ and $L^H$ the corresponding fixed field. Let $\sigma _1,\dots,\sigma _n\in H$ such that $\sigma _1(r-w_{\mathfrak{s}}), \dots,\sigma _n(r-w_{\mathfrak{s}})$ are the roots of the minimal polynomial of $r-w_{\mathfrak{s}}$ over $L^H$ . Hence $\sigma _i(r)\in \mathfrak{R}$ and $\sigma _i(r)\neq \sigma _j(r)$ for any $i,j=1,\dots,n$ , $i\neq j$ . From

\begin{align*} \prod _{i=1}^n\sigma _i(r-w_{\mathfrak{s}})\in L^H\qquad \text{and} \qquad v\bigg (\prod _{i=1}^n\sigma _i(r-w_{\mathfrak{s}})\bigg )=n\cdot \rho _{\mathfrak{s}}, \end{align*}

it follows that $|\rho _{\mathfrak{s}}|_p\mid n$ , and so $|\rho _{\mathfrak{s}}|_p\leq n$ , since $L^H/K$ is tamely ramified. By definition of $H$ (see for example [17, Definition 9.3]) we have

\begin{align*} v\big (\tfrac{\sigma _i(r-w_{\mathfrak{s}})}{r-w_{\mathfrak{s}}}-1\big )\gt 0,\quad \text{ and so }\quad v\big (\sigma _i(r)-r\big )=v\big (\sigma _i(r-w_{\mathfrak{s}})-(r-w_{\mathfrak{s}})\big )\gt \rho _{\mathfrak{s}} \end{align*}

for any $i=1,\dots,n$ . Therefore $\sigma _i(r)\in \mathfrak{s}^{\prime}$ for all $i$ and so $n\leq |\mathfrak{s}^{\prime}|$ . Thus $n=|\mathfrak{s}^{\prime}|=|\rho _{\mathfrak{s}}|_p$ and $\mathfrak{s}\subseteq \mathfrak{s}^{\prime}=\{\sigma _i(r)\mid i=1,\dots,n\}$ . Finally, as $\mathfrak{s}^{\prime}$ contains only conjugates of $r\in \mathfrak{s}$ , the cluster $\mathfrak{s}^{\prime}$ is union of orbits of $\mathfrak{s}$ . In particular, $|\mathfrak{s}|\mid |\mathfrak{s}^{\prime}|=|\rho _{\mathfrak{s}}|_p$ , and so $|\mathfrak{s}|$ is a $p$ -power. The rest of the Lemma follows.

Proposition 3.31. The polynomial $f$ has an almost rational cluster picture if and only if for every proper cluster $\mathfrak{s}\in \Sigma _f$ one of the following is satisfied:

1. (a) the smallest disc containing $\mathfrak{s}$ also contains a rational point;

2. (b) $p\gt 0$ and after a translation by an element of $K$ , the elements in $\mathfrak{s}$ are all the roots of a monic polynomial with coefficients in $K^{\textrm{s}}$ of $p$ -power degree and constant term $c$ such that $|v(c)|_p\geq 1$ .

Proof. First of all, note that point (a) is equivalent to requiring $d_{\mathfrak{s}}=\rho _{\mathfrak{s}}$ . Therefore, by Lemma 3.30 it only remains to show that if $\mathfrak{s}\in \Sigma _f$ with $d_{\mathfrak{s}}\gt \rho _{\mathfrak{s}}$ and (b) is satisfied, then $|\mathfrak{s}|\leq |\rho _{\mathfrak{s}}|_p$ . Let $F\in K^{\textrm{s}}[x]$ be the polynomial in (b) and let $w\in K$ such that $r-w$ , for $r\in \mathfrak{s}$ , are all the roots of $F$ . We have $\rho _{\mathfrak{s}}\geq \min _{r\in \mathfrak{s}}v(r-w)$ . Fix $r\in \mathfrak{s}$ such that $\rho _{\mathfrak{s}}\geq v(r-w)\;=\!:\;\rho$ . Since $d_{\mathfrak{s}}\gt \rho _{\mathfrak{s}}\geq v(r-w)$ , we have $v(r'-w)=v(r-w)=\rho$ for any $r'\in \mathfrak{s}$ . Then

\begin{align*} |\mathfrak{s}|=\deg F = |1/\deg F|_p\leq |v(c)/\deg F|_p=|\rho |_p. \end{align*}

We will prove that $\rho =\rho _{\mathfrak{s}}$ , so that $|\mathfrak{s}|\leq |\rho |_p=|\rho _{\mathfrak{s}}|_p$ , as required. We already know that $\rho _{\mathfrak{s}}\geq \rho$ . Suppose by contradiction that $\rho _{\mathfrak{s}}\gt \rho$ . Let $w_{\mathfrak{s}}$ be a rational centre of $\mathfrak{s}$ and let $r_{\mathfrak{s}}\in \mathfrak{s}$ such that $v(r_{\mathfrak{s}}-w_{\mathfrak{s}})=\rho _{\mathfrak{s}}$ . Hence

\begin{align*} v(w-w_{\mathfrak{s}})=v(w-r_{\mathfrak{s}}+r_{\mathfrak{s}}-w_{\mathfrak{s}})=\min \{\rho,\rho _{\mathfrak{s}}\}=\rho. \end{align*}

But then $\rho \in \mathbb{Z}$ , which contradicts $|\mathfrak{s}|\leq |\rho |_p$ .

Example 3.32. Let $p$ be a prime number and let $a\in \mathbb{Z}_p$ , $b\in \mathbb{Z}_p^\times$ such that the polynomial $x^2+ax+b$ is not a square modulo $p$ . Let $f\in \mathbb{Q}_p[x]$ given by $f(x)=(x^6+ap^4x^3+bp^8)((x-p)^3-p^{11})$ . For any prime $p$ the rational cluster picture of $f$ is

where $\rho _{\mathfrak{t}_3}=\frac{4}{3}$ , $\rho _{\mathfrak{t}_4}=\frac{11}{3}$ , and $\rho _{\mathfrak{R}}=1$ .

If $p\neq 3$ , then the proper clusters of $\Sigma _f$ coincide with the rational clusters above and $d_{\mathfrak{s}}=\rho _{\mathfrak{s}}$ for any $\mathfrak{s}=\mathfrak{t}_3,\mathfrak{t}_4,\mathfrak{R}$ . In particular, $f$ has an almost rational cluster picture when $p\neq 3$ .

Suppose $p=3$ . Then the cluster picture of $f$ is

where $d_{\mathfrak{t}_1}=d_{\mathfrak{t}_2}=\frac{11}{6}$ , $d_{\mathfrak{t}_3}=\rho _{\mathfrak{t}_1}=\rho _{\mathfrak{t}_2}=\frac{4}{3}$ , $d_{\mathfrak{t}_4}=\frac{25}{6}$ and $d_{\mathfrak{R}}=1$ . Thus $f$ has an almost rational cluster picture for all $p$ .

We conclude this section by showing that the cluster picture centred at $w\in K$ completely determines the Newton polygon of the translation of $f$ by $w$ .

Definition 3.33. Let $z\in \bar{K}$ . A cluster centred at $z$ is a cluster cut out by a $v$ -adic disc of the form $\mathcal{D}=\{x\in \bar{K}\mid v(x-z)\geq d\}$ for some $d\in \mathbb{Q}$ .

Definition 3.34. Let $z\in \bar{K}$ . Define $\Sigma _f^z$ to be the set of all clusters centred at $z$ . Write $\Sigma _f^z$ for the set $\Sigma _f^z\smallsetminus \{\{z\}\}$ . Note that $\Sigma _f^z$ is nested, that is every cluster $\mathfrak{s}\in \Sigma _f^z$ has at most one child in $\Sigma _f^z$ .

Definition 3.35. Let $z\in \bar{K}$ , and let $\mathfrak{s}\in \Sigma _f\setminus \{\{z\}\}$ . The radius of $\mathfrak{s}$ with respect to the centre $z$ is

\begin{align*} \rho _{\mathfrak{s}}^z=\min _{r\in \mathfrak{s}}v(r-z). \end{align*}

The cluster picture centred at $z$ of $f$ is the collection of all clusters in $\Sigma _f^z$ together with their radii with respect to $z$ . Finally set

\begin{align*} \epsilon _{\mathfrak{s}}^z\;:\!=\;v(c_f)+\sum _{r\in \mathfrak{R}}\rho _{r\wedge \mathfrak{s}}^z. \end{align*}

Remark 3.36. From the definitions above, if $\mathfrak{s}$ is a cluster centred at $z\in K^{\textrm{s}}$ , then $\mathfrak{s}=\mathfrak{R}\cap \{x\in \bar{K}\mid v(x-z)\geq \rho _{\mathfrak{s}}^z\}$ . But this does not mean $z$ is a centre for $\mathfrak{s}$ , that is false in general. For example, $\mathfrak{R}$ is clearly a cluster centred at any $z\in K^{\textrm{s}}$ , but there are elements of $K^{\textrm{s}}$ which are not centres of $\mathfrak{R}$ , for example any $z\in K^{\textrm{s}}$ with valuation $v(z)\lt \min _{r\in \mathfrak{R}}v(r)$ .

Remark 3.37. Let $\mathfrak{s}\in \Sigma _f$ be a proper cluster with centre $z$ and rational centre $w$ . Then $\mathfrak{s}\in \Sigma _f^{z}$ , $d_{\mathfrak{s}}=\rho ^z_{\mathfrak{s}}$ , $\nu _{\mathfrak{s}}=\epsilon ^z_{\mathfrak{s}}$ , $\rho _{\mathfrak{s}}=\rho _{\mathfrak{s}}^w$ , and $\epsilon _{\mathfrak{s}}=\epsilon _{\mathfrak{s}}^w$ . Furthermore, $\mathfrak{s}\in \Sigma _f^{\textrm{rat}}$ if and only if $\mathfrak{s}\in \Sigma _f^{w}$ .

The following result gives a complete description of the Newton polygon of the translation of $f$ by $w\in K$ , knowing the cluster picture centred at $w$ of $f$ .

Lemma 3.38. Let $w\in K$ and let $f_w(x)=f(x+w)$ . Then there is a $1$ -to- $1$ correspondence between the clusters in $\Sigma _f^w$ and the edges of $\texttt{NP}(f_w)$ . More explicitly, let ${\mathfrak{s}}_1\subset \dots \subset{\mathfrak{s}}_n=\mathfrak{R}$ be the clusters in $\Sigma _f^w$ and let ${\mathfrak{s}}_0=\{w\}$ if $\{w\}\in \Sigma _f^w$ or ${\mathfrak{s}}_0=\varnothing$ otherwise. Then $\texttt{NP}(f_w)$ has vertices $Q_i$ , $i=0,\dots,n$ , where

• $Q_n=(|\mathfrak{R}|,\epsilon _{\mathfrak{R}}^w-|\mathfrak{R}|\rho _{\mathfrak{R}}^w)=(\deg f,v(c_f))$ ,

• $Q_i=(|{\mathfrak{s}}_i|,\epsilon _{{\mathfrak{s}}_i}^w-|{\mathfrak{s}}_i|\rho _{{\mathfrak{s}}_i}^w)=(|{\mathfrak{s}}_i|,\epsilon _{{\mathfrak{s}}_{i+1}}^w-|{\mathfrak{s}}_i|\rho _{{\mathfrak{s}}_{i+1}}^w)$ , for $i=1,\dots,n-1$ ,

• $Q_0=(|{\mathfrak{s}}_0|,\epsilon _{{\mathfrak{s}}_{1}}^w-|{\mathfrak{s}}_0|\rho _{{\mathfrak{s}}_{1}}^w)$ .

and edges $L_i$ , $i=1,\dots,n$ , of slope $-\rho _{{\mathfrak{s}}_i}^w$ linking $Q_{i-1}$ and $Q_i$ .

Furthermore, for any $i=1,\dots,n$ we have

\begin{align*} \overline{f_w|_{L_i}}(x^{b_i})= \tfrac{u}{\pi ^{v(u)}}\textstyle \prod _{r\in{\mathfrak{s}}_i\setminus{\mathfrak{s}}_{i-1}} (x+\tfrac{w-r}{\pi ^{\rho _i}})\mod \pi,\qquad u=c_f\textstyle \prod _{r\in \mathfrak{R}\setminus \mathfrak{s}} (w-r), \end{align*}

where $\rho _i=\rho _{{\mathfrak{s}}_i}^w$ , and $b_i$ is the denominator of $\rho _i$ .

Proof. Without loss of generality we can assume $w=0$ so that $f_w=f$ . First note that the coordinates of $Q_n$ are trivial. Now consider a factorisation $f=c_f\cdot g_0\cdot g_1\cdots g_s$ of Theorem 2.2. Recall the polynomials $g_j$ are monic and $g_0\mid x$ . Let $\mathfrak{R}_j$ be the set of roots of $g_j$ . It follows from the definition of cluster centred at $0$ that

\begin{align*} n=s,\quad \mbox{and}\quad{\mathfrak{s}}_i=\bigcup _{j=0}^i\mathfrak{R}_j\quad \text{for all }i=0,\dots,n. \end{align*}

Therefore ${\mathfrak{s}}_0=\mathfrak{R}_0$ and $\mathfrak{R}_i={\mathfrak{s}}_i\setminus{\mathfrak{s}}_{i-1}$ for any $i=1,\dots,n$ .

Let $i=1,\dots,n-1$ . Then the $x$ -coordinate of $Q_i$ follows as

\begin{align*} |{\mathfrak{s}}_i|=\sum _{j=0}^i|\mathfrak{R}_j|=\sum _{j=0}^i\deg g_j=\deg \prod _{j=0}^ig_j. \end{align*}

The $y$ -coordinate of $Q_i$ equals the sum of $v(c_f)$ and the valuation of the constant term of $\prod _{j=i+1}^n g_j$ , so

\begin{align*} Q_i=\bigg (|{\mathfrak{s}}_i|,v(c_f)+\sum _{j=i+1}^n|\mathfrak{R}_j|v(r_j)\bigg ), \end{align*}

where $r_j$ is any root in $\mathfrak{R}_j$ . But since ${\mathfrak{s}}_i=\bigcup _{j=0}^i\mathfrak{R}_j$ , we have $v(r_j)=\rho _{{\mathfrak{s}}_j}^0$ . Therefore

\begin{align*} v(c_f)+\sum _{j=i+1}^n|\mathfrak{R}_j|v(r_j)=v(c_f)+\sum _{j=i+1}^n(|{\mathfrak{s}}_j|-|{\mathfrak{s}}_{j-1}|)\rho _{{\mathfrak{s}}_j}^0=\epsilon _{{\mathfrak{s}}_i}^0-|{\mathfrak{s}}_i|\rho _{{\mathfrak{s}}_i}^0. \end{align*}

Moreover,

\begin{align*} \epsilon _{{\mathfrak{s}}_i}^0-|{\mathfrak{s}}_i|\rho _{{\mathfrak{s}}_i}^0=\epsilon _{{\mathfrak{s}}_{i+1}}^0-|{\mathfrak{s}}_i|\rho _{{\mathfrak{s}}_{i+1}}^0 \end{align*}

from the easy computation $\epsilon _{{\mathfrak{s}}_i}^0-\epsilon _{{\mathfrak{s}}_{i+1}}^0=|{\mathfrak{s}}_i|\big (\rho _{{\mathfrak{s}}_i}^0-\rho _{{\mathfrak{s}}_{i+1}}^0\big ).$ Finally the $x$ -coordinate of $Q_0$ is trivial, while its $y$ -coordinate equals

\begin{align*} v(c_f)+\sum _{j=1}^n|\mathfrak{R}_j|v(r_j)=v(c_f)+\sum _{j=1}^n(|{\mathfrak{s}}_j|-|{\mathfrak{s}}_{j-1}|)\rho _{{\mathfrak{s}}_j}^0=\epsilon _{{\mathfrak{s}}_1}^0-|{\mathfrak{s}}_0|\rho _{{\mathfrak{s}}_1}^0, \end{align*}

that concludes the first part of the proof as $|{\mathfrak{s}}_0|=|\mathfrak{R}_0|=\deg g_0$ .

The computation of $f|_{L_i}$ follows from Remark 2.7. Indeed, let $i=1,\dots,n$ , and define $\bar{c}_i=u/\pi ^{v(u)}\mod \pi$ , where $u=c_f\textstyle \prod _{j=i+1}^ng_j(0)$ . Then ${\overline{f|_{L_i}}}(x^{b_i})=\bar{c}_i\cdot{\overline{g_i|_{\texttt{NP}(g_i)}}}(x^{b_i})$ , where $b_i$ is the denominator of $\rho _{{\mathfrak{s}}_i}^0$ . But

\begin{align*}{\overline{g_i|_{\texttt{NP}(g_i)}}}(x^{b_i})=g_i\big (\pi ^{\rho _{{\mathfrak{s}}_i}^0}x\big )/\pi ^{\rho _{{\mathfrak{s}}_i}^0\deg g_i}\mod \pi. \end{align*}

Thus the Lemma follows as $\mathfrak{R}_i={\mathfrak{s}}_i\setminus{\mathfrak{s}}_{i-1}$ .

Notation 3.39. Let $\mathfrak{s}\in \Sigma _f^w$ . Following the notation of Lemma 3.38, let $i\in \{1,\dots,n\}$ be such that $\mathfrak{s}={\mathfrak{s}}_i$ . We will write $L_{\mathfrak{s}}^w$ for the edge $L_i$ .

4. Description of a regular model

From now on, assume $\textrm{char}(K)\neq 2$ and let $C/K$ be a hyperelliptic curve, that is a geometrically connected, smooth, projective curve, equipped with a separable morphism $C\rightarrow{\mathbb{P}}^1_K$ of degree $2$ . Let $y^2=f(x)$ be a Weierstrass equation of $C$ . Suppose $\deg f\gt 1$ . Let $g$ be the genus of $C$ . Accordingly with [14] we define the cluster picture of $C$ as the cluster picture of $f$ . Analogously, all definitions and notations attached to $f$ given in Section 3 (e.g. $\Sigma _f$ , $\Sigma _f^{\textrm{rat}}$ , $\Sigma _f^z$ ) are given for $C$ in the same way (e.g. $\Sigma _C$ , $\Sigma _C^{\textrm{rat}}$ , $\Sigma _C^z$ ). In particular, we will say that $C$ has an almost rational cluster picture if $f$ does (Definition 3.26).

For the following sections we will use the main definitions, notations and results of [1, § $3$ ]. In particular, we recall (without stating) the definitions of Newton polytopes $\Delta$ and $\Delta _v$ attached to a polynomial $g\in K[x,y]$ , $v$ -vertices/edges/faces of $\Delta$ , the denominator $\delta _\lambda$ of a $v$ -face/edge $\lambda$ , the slopes $s_1^\lambda, s_2^\lambda$ of a $v$ -edge $\lambda$ .

Notation 4.1. We denote by $\Delta _v^w$ and $\Delta ^w$ respectively the polytopes $\Delta _v$ and $\Delta$ attached to the polynomial $g_w(x,y)=y^2-f(x+w)$ . The piecewise affine function $v\;:\;\Delta ^w\rightarrow{\mathbb{R}}$ determining the bijection $\Delta ^w\rightarrow \Delta _v^w$ , $P\mapsto (P,v(P))$ , will be denoted by $v$ (with a little abuse of notation). For a $v$ -face $F$ of $\Delta ^w$ , denote by $v_F\;:\;\Delta ^w\rightarrow{\mathbb{R}}$ the linear function equal to $v$ on $F$ . Since the projection $\Delta _v^w\rightarrow \Delta ^w$ is a bijection, given a vertex/edge/face $\lambda$ of $\Delta _v^w$ we will denote by the same symbol $\lambda$ the corresponding $v$ -vertex/edge/face of $\Delta ^w$ . Since they are mainly used for indexing, this will not cause confusion.

Notation 4.2. Given a $v$ -edge $\lambda$ of $\Delta ^w$ , we will denote by $r_\lambda$ the smallest non-negative integer such that we can fix $\frac{n_i}{d_i}\in \mathbb{Q}$ , for $i=0,\dots,r_\lambda +1$ so that

\begin{align*} s_1^\lambda =\frac{n_0}{d_0}\gt \frac{n_1}{d_1}\gt \ldots \gt \frac{n_{r_\lambda }}{d_{r_\lambda }}\gt \frac{n_{r_\lambda +1}}{d_{r_\lambda +1}}=s_2^\lambda,\quad \text{with} \quad \begin{vmatrix}n_i\!\!\!&n_{i+1}\\ d_i\!\!\!&d_{i+1}\\\end{vmatrix}=1. \end{align*}

Thanks to Lemma 3.38 we can explicitly relate the Newton polytope $\Delta _v^w$ of $g_w(x,y)$ and the cluster picture centred at $w$ of $C$ .

Lemma 4.3. Let $w\in K$ . Then there is a $1$ -to- $1$ correspondence between the clusters in $\Sigma _C^w$ and the faces of the Newton polytope $\Delta _v^w$ . More explicitly, let ${\mathfrak{s}}_1\subset \dots \subset{\mathfrak{s}}_n=\mathfrak{R}$ be the clusters in $\Sigma _C^w$ and let ${\mathfrak{s}}_0=\{w\}$ if $\{w\}\in \Sigma _C^w$ or ${\mathfrak{s}}_0=\varnothing$ otherwise. Then $\Delta _v^w$ has vertices $T$ , $Q_i$ , $i=0,\dots,n$ , where

• $T=(0,2,0)$ ,

• $Q_n=(|\mathfrak{R}|,0,v(c_f))$ ,

• $Q_i=(|{\mathfrak{s}}_i|,0,\epsilon _{{\mathfrak{s}}_{i+1}}^w-|{\mathfrak{s}}_i|\rho _{{\mathfrak{s}}_{i+1}}^w)$ for $i=0,\dots,n-1$ ,

and edges $L_i$ ( $i=1,\dots,n$ ), linking $Q_{i-1}$ and $Q_i$ , and $V_j$ ( $j=0,\dots,n$ ), linking $Q_j$ and $T$ . Furthermore, (possible choices for) the slopes of the $v$ -edges of $\Delta ^w$ are:

• \begin{align*} s_1^{V_n}=\delta _{V_n}\tfrac{-v(c_f)+(|\mathfrak{R}|-2g)\rho _{\mathfrak{R}}^w}{2}\quad \mbox{and}\quad s_2^{V_n}=\lfloor s_1^{V_n}-1\rfloor ; \end{align*}

• \begin{align*} \renewcommand{\arraystretch }{2} \begin{array}{l} s_1^{V_i}=\delta _{V_i}\left (-\frac{\epsilon _{{\mathfrak{s}}_{i}}^w}{2}+\left (\left \lfloor \frac{|{\mathfrak{s}}_i|}{2}\right \rfloor +1\right )\rho _{{\mathfrak{s}}_{i}}^w\right ),\\ s_2^{V_i}=\delta _{V_i}\left (-\frac{\epsilon _{{\mathfrak{s}}_{i+1}}^w}{2}+\left (\left \lfloor \frac{|{\mathfrak{s}}_i|}{2}\right \rfloor +1\right )\rho _{{\mathfrak{s}}_{i+1}}^w\right ) \end{array}\renewcommand{\arraystretch }{1} \quad \text{for all }i=1,\dots,n-1; \end{align*}

• \begin{align*} s_1^{V_0}=\delta _{V_0}\left (\tfrac{\epsilon _{{\mathfrak{s}}_1}^w}{2}-\rho _{{\mathfrak{s}}_1}^w\right )\quad \text{and}\quad s_2^{V_0}=\lfloor s_1^{V_0}-1\rfloor ; \end{align*}

• \begin{align*} s_1^{L_i}=\delta _{L_i}\left (-\tfrac{\epsilon _{{\mathfrak{s}}_{i}}^w}{2}+\left (\left \lfloor \tfrac{|{\mathfrak{s}}_i|}{2}\right \rfloor +1\right )\rho _{{\mathfrak{s}}_{i}}^w\right ) \quad \text{and}\quad s_2^{L_i}=\lfloor s_1^{L_i}-1\rfloor, \end{align*}

  for all $i=1,\dots,n$ . In particular, as $\delta _{L_i}$ is the denominator of $\rho _{{\mathfrak{s}}_{i}}^w$ ,

  \begin{align*} r_{L_i}=\begin{cases}1&\mbox{if }\delta _{L_i}\epsilon _{{\mathfrak{s}}_i}^w\mbox{ is odd,}\\0&\mbox{if }\delta _{L_i}\epsilon _{{\mathfrak{s}}_i}^w\mbox{ is even.}\end{cases} \end{align*}

Finally, for suitable choices of basis of the lattices in [ 1, 3.4, 3.5], we have

\begin{align*} \overline{g_w|_{L_i}}(x^{b_i})= -\tfrac{u}{\pi ^{v(u)}}\textstyle \prod _{r\in{\mathfrak{s}}_i\setminus{\mathfrak{s}}_{i-1}} (x+\tfrac{w-r}{\pi ^{\rho _i}})\mod \pi,\qquad u=c_f\textstyle \prod _{r\in \mathfrak{R}\setminus{\mathfrak{s}}_i} (w-r), \end{align*}

for any $i=1,\dots,n$ , where $\rho _i=\rho _{{\mathfrak{s}}_i}^w$ , and $b_i$ is the denominator of $\rho _i$ ;

\begin{align*} \overline{g_w|_{V_j}}(y)=y^{|\bar{V}_j(\mathbb{Z})_{\mathbb{Z}}|-1}- \tfrac{u}{\pi ^{v(u)}}\mod \pi,\qquad u=c_f\textstyle \prod _{r\in \mathfrak{R}\setminus{\mathfrak{s}}_j} (w-r), \end{align*}

for any $j=0,\dots,n$ , where $|\bar{V}_j(\mathbb{Z})_{\mathbb{Z}}|$ is the number of integer points $P$ on the $v$ -edge $V_j$ with $v(P)\in \mathbb{Z}$ , endpoints included.

Proof. The structure of $\Delta _v^w$ follows from Lemma 3.38. For the computation of the slopes, we only need to individuate, for all the $v$ -edges, the two points $P_0$ and $P_1$ of [1, Definition 3.12]. It is easy to see that the followings are admissible choices.

• For $V_i$ and $L_i$ ( $i=1,\dots,n$ ), choose $P_0=(|{\mathfrak{s}}_i|,0)$ and $P_1=\left (\left \lfloor \frac{|{\mathfrak{s}}_i|-1}{2}\right \rfloor,1\right )$ .

• For $V_0$ , choose $P_0=(0,2)$ and $P_1=(1,1)$ ;

The second part of the Lemma then follows from the first one. The computations of the reductions also follows from Lemma 3.38 by choosing the lattices $Q_{i-1}+(b_i,0)\mathbb{Z}$ for $g_w|_{L_i}$ and $Q_{i}+({-}|{\mathfrak{s}}_i|/a,2/a)\mathbb{Z}$ for $g_w|_{V_j}$ , where $a=|\bar{V}_j(\mathbb{Z})_{\mathbb{Z}}|-1$ .

Notation 4.4. Let $C$ be as above and let $w\in K$ . For every cluster $\mathfrak{s}\in \Sigma _C^w$ denote by $F_{\mathfrak{s}}^w$ the $v$ -face of the Newton polytope $\Delta ^w$ of $g_w(x,y)=y^2-f(x+w)$ that corresponds to $\mathfrak{s}$ .

Following the notation of Lemma 4.3, let $i\in \{1,\dots,n\}$ be such that $\mathfrak{s}={\mathfrak{s}}_i$ . We will write $L_{\mathfrak{s}}^w$ , $V_{\mathfrak{s}}^w$ , $V_0^w$ for the $v$ -edges $L_i$ , $V_i$ , $V_0$ , respectively.

Example 4.5. Let $C$ be the hyperelliptic curve over $\mathbb{Q}_3$ given by the equation $y^2=f(x)$ where $f(x)=x^{11}-3x^6+9x^5-27$ is the polynomial of Example 3.20 .

Its cluster picture centred at $0$ is

where the subscripts represent the radii with respect to $0$ . As we can see, $\Sigma _C^0$ consists of two clusters: ${\mathfrak{s}}_1$ of size $6$ , radius $\frac{1}{3}$ and $\epsilon _{{\mathfrak{s}}_1}^0=3$ , and ${\mathfrak{s}}_2=\mathfrak{R}$ of size $11$ , radius $\frac{1}{5}$ and $\epsilon _{{\mathfrak{s}}_2}^0=\frac{11}{5}$ . Therefore the picture of $\Delta ^0$ broken into $v$ -faces will be

where $T=(0,2)$ , $Q_0=(0,0)$ , $Q_1=(6,0)$ , and $Q_2=(11,0)$ . Denoting the values of $v$ on vertices, the picture becomes

To state the theorems which describe the special fibre of the proper flat model $\mathcal{C}$ of $C$ we will construct in Section 5, we need some definitions.

Definition 4.6. Let $F/K$ be an unramified extension and let $\Sigma _F=\Sigma _{C_{F}}^{\textrm{rat}}$ (i.e. set of clusters cut out by discs with centre in $F$ ). For any proper $\mathfrak{s}\in \Sigma _F$ let $G_{\mathfrak{s}}=\textrm{Stab}_{G_K}(\mathfrak{s})$ and $K_{\mathfrak{s}}=\left ( K^{\textrm{s}}\right )^{G_{\mathfrak{s}}}$ . We define the following quantities:

$\mathfrak{s}\in \Sigma _F$ , proper
radius	$\rho _{\mathfrak{s}}$	$\max _{w\in F}\min _{r\in \mathfrak{s}} v(r-w)$
	$b_{\mathfrak{s}}$	denominator of $\rho _{\mathfrak{s}}$
	$\epsilon _{\mathfrak{s}}$	$v(c_f) + \sum _{r\in \mathfrak{R}} \rho _{r\wedge \mathfrak{s}}$
	$D_{\mathfrak{s}}$	$1$ if $b_{\mathfrak{s}}\epsilon _{\mathfrak{s}}$ odd, $2$ if $b_{\mathfrak{s}}\epsilon _{\mathfrak{s}}$ even
multiplicity	$m_{\mathfrak{s}}$	$(3-D_{\mathfrak{s}})b_{\mathfrak{s}}$
parity	$p_{\mathfrak{s}}$	$1$ if $|\mathfrak{s}|$ is odd, $2$ if $|\mathfrak{s}|$ is even
slope	$s_{\mathfrak{s}}$	$\frac 12(|\mathfrak{s}|\rho _{\mathfrak{s}}+p_{\mathfrak{s}}\rho _{\mathfrak{s}}-\epsilon _{\mathfrak{s}})$
	$\gamma _{\mathfrak{s}}$	$2$ if $\mathfrak{s}$ is even and $\epsilon _{\mathfrak{s}}\!-\!|\mathfrak{s}|\rho _{\mathfrak{s}}$ is odd, $1$ otherwise
	$p_{\mathfrak{s}}^0$	$1$ if $\mathfrak{s}$ is minimal and $\mathfrak{s}\cap K_{\mathfrak{s}}\neq \varnothing$ , $2$ otherwise
	$ {s}^0_{\mathfrak{s}} $	$ -\epsilon_{\mathfrak{s}}/2 + \rho_{\mathfrak{s}} $
	$\gamma _{\mathfrak{s}}^0$	2 if $p_{\mathfrak{s}}^0=2$ and $\epsilon _{\mathfrak{s}}$ is odd, 1 otherwise

Lemma 4.7. Keep the notation of the previous definition and let $\mathfrak{s}\in \Sigma _K$ . Then $\mathfrak{s}\in \Sigma _F$ but the quantities in Definition 4.6 do not depend on $F$ .

Proof. A cluster $\mathfrak{s}\in \Sigma _F$ belongs to $\Sigma _K$ if and only if $\sigma (\mathfrak{s})=\mathfrak{s}$ for any $\sigma \in G_K$ . Then the result follows from Lemma A.1.

Remark 4.8. Lemma 4.3 shed some light on the quantities we defined in Definition 4.6 . Let $\mathfrak{s}\in \Sigma _F$ . Fix a rational centre $w_{\mathfrak{s}}\in F$ of $\mathfrak{s}$ such that $w_{\mathfrak{s}}\in K_{\mathfrak{s}}$ if $p_{\mathfrak{s}}^0=1$ . Denoting $V=V_{\mathfrak{s}}^{w_{\mathfrak{s}}}$ , $L=L_{\mathfrak{s}}^{w_{\mathfrak{s}}}$ , and $V_0=V_0^{w_{\mathfrak{s}}}$ , we have:

• $b_{\mathfrak{s}}=\delta _L$ and $r_L=2-D_{\mathfrak{s}}$ .

• $\gamma _{\mathfrak{s}}=\delta _V$ , $p_{\mathfrak{s}}/\gamma _{\mathfrak{s}}=\bar{V}(\mathbb{Z})_{\mathbb{Z}}-1$ and $s_1^V=\gamma _{\mathfrak{s}} s_{\mathfrak{s}}$ . If $V$ is internal, that is $\mathfrak{s}\neq \mathfrak{R}$ , then $s_2^V=\gamma _{\mathfrak{s}}(s_{\mathfrak{s}}-p_{\mathfrak{s}}\tfrac{\rho _{\mathfrak{s}}-\rho _{P(\mathfrak{s})}}{2})$ .

• If $\mathfrak{s}$ is minimal and so $V_0$ is an edge of $F_{\mathfrak{s}}^{w_{\mathfrak{s}}}$ , then $\gamma _{\mathfrak{s}}^0=\delta _{V_0}$ , $p_{\mathfrak{s}}^0/\gamma _{\mathfrak{s}}^0=\bar{V}_0(\mathbb{Z})_{\mathbb{Z}}-1$ and $s_1^{V_0}=-\gamma _{\mathfrak{s}}^0 s_{\mathfrak{s}}^0$ .

Lemma 4.9. Let $\mathfrak{s}\in \Sigma _C^{\textrm{rat}}$ with rational centre $w\in K$ . Then $D_{\mathfrak{s}}=1$ if and only if $v_{F_{\mathfrak{s}}^w}((a,1))\notin \mathbb{Z}$ , for every $a\in \mathbb{Z}$ .

Proof. If $D_{\mathfrak{s}}=1$ then $r_{L_{\mathfrak{s}}^w}=1$ by Lemma 4.3, and so $v_{F_{\mathfrak{s}}^w}((a,1))\notin \mathbb{Z}$ , for every $a\in \mathbb{Z}$ . Now let $c,d\in \mathbb{Z}$ such that $\rho _{\mathfrak{s}}\cdot c+d=1/b_{\mathfrak{s}}$ . If $D_{\mathfrak{s}}=2$ , then $b_{\mathfrak{s}}\epsilon _{\mathfrak{s}}\in 2\mathbb{Z}$ , so

\begin{align*} v_{F_{\mathfrak{s}}^w}(cb_{\mathfrak{s}}\epsilon _{\mathfrak{s}}/2,1)=\frac{v_{F_{\mathfrak{s}}^w}((cb_{\mathfrak{s}}\epsilon _{\mathfrak{s}},0))}{2}=\frac{\epsilon _{\mathfrak{s}}-(cb_{\mathfrak{s}}\epsilon _{\mathfrak{s}})\rho _{\mathfrak{s}}}{2}=\frac{db_{\mathfrak{s}}\epsilon _{\mathfrak{s}}}{2}\in \mathbb{Z}, \end{align*}

as required.

Definition 4.10. We say that $C$ is $y$ -regular if $p\nmid D_{\mathfrak{s}}$ for every proper $\mathfrak{s}\in \Sigma _C^{\textrm{rat}}$ , that is if either $p\neq 2$ or $D_{\mathfrak{s}}=1$ for any proper $\mathfrak{s}\in \Sigma _C^{\textrm{rat}}$ .

Remark 4.11. Let $F/K$ be an unramified extension. Then from Lemma 4.7 , if $C_F$ is $y$ -regular then $C$ is $y$ -regular.

Lemma 4.12. The hyperelliptic curve $C$ is $\Delta _v$ -regular if and only if $C$ is $y$ -regular and $f$ is $\texttt{NP}$ -regular.

Proof. Let $g(x,y)=y^2-f(x)$ . If $C$ is $y$ -regular and $f$ is $\texttt{NP}$ -regular, then $C$ is $\Delta _v$ -regular by Lemma 4.3 and Lemma 4.9.

Conversely, if $C$ is $\Delta _v$ -regular, then $f$ is $\texttt{NP}$ -regular, and all clusters have rational centre $0$ by Corollary 3.25. It remains to show that if $p=2$ then $D_{\mathfrak{s}}=1$ for every proper $\mathfrak{s}\in \Sigma _C^{\textrm{rat}}$ . Suppose there exists $\mathfrak{s}\in \Sigma _C^{\textrm{rat}}$ such that $D_{\mathfrak{s}}=2$ . Consider the variety $\bar{X}_{F_{\mathfrak{s}}^0}$ ([1, Definition 3.7]). By Lemma 4.9, the smoothness of $\bar{X}_{F_{\mathfrak{s}}^0}$ implies there exists $\mathfrak{s}^{\prime}\in \Sigma _C^{\textrm{rat}}$ , such that $|\mathfrak{s}|-|\mathfrak{s}^{\prime}|=1$ . Hence $\rho _{\mathfrak{s}}\in \mathbb{Z}$ from Lemma 3.12. Therefore $\bar{F}_{\mathfrak{s}}^0(\mathbb{Z})=\bar{F}_{\mathfrak{s}}^0(\mathbb{Z})_{\mathbb{Z}}$ , by Lemma 4.9. But this gives a contradiction as it forces either $\overline{g|_{V_{\mathfrak{s}^{\prime}}^0}}$ or $\overline{g|_{V_{\mathfrak{s}}^0}}$ to be a square.

Definition 4.13. Let $\mathfrak{s}\in \Sigma _F$ be a proper cluster and let $c\in \{0,\dots,b_{\mathfrak{s}}-1\}$ such that $c\rho _{\mathfrak{s}}-\frac{1}{b_{\mathfrak{s}}}\in \mathbb{Z}$ . Define

\begin{align*} \tilde{\mathfrak{s}}=\{\mathfrak{s}^{\prime}\in \Sigma _F\cup \{\varnothing \}\mid \mathfrak{s}^{\prime}\lt \mathfrak{s}\text{ and }\tfrac{|\mathfrak{s}^{\prime}|}{b_{\mathfrak{s}}}-c\epsilon _{\mathfrak{s}}\notin 2\mathbb{Z}\}, \end{align*}

where $\varnothing \lt \mathfrak{s}$ if $\mathfrak{s}$ is minimal and $p_{\mathfrak{s}}^0=2$ .

The genus $g(\mathfrak{s})$ of a rational cluster $\mathfrak{s}\in \Sigma _F$ is defined as follows:

• If $D_{\mathfrak{s}}=1$ , then $g(\mathfrak{s})=0$ .

• If $D_{\mathfrak{s}}=2$ , then $2g(\mathfrak{s})+1$ or $2g(\mathfrak{s})+2$ equals

  \begin{align*} \frac{|\mathfrak{s}|-\sum _{\mathfrak{s}^{\prime}\in \Sigma _F,\mathfrak{s}^{\prime}\lt \mathfrak{s}}|\mathfrak{s}^{\prime}|}{b_{\mathfrak{s}}}+|\tilde{\mathfrak{s}}|. \end{align*}

Definition 4.14. Let $\Sigma _C^{\textrm{min}}$ be the set of rationally minimal clusters of $C$ and let $\Sigma \subseteq \Sigma _C^{\textrm{min}}$ non-empty. For each cluster $\mathfrak{s}\in \Sigma$ , fix a rational centre $w_{\mathfrak{s}}$ ; if possible, choose $w_{\mathfrak{s}}\in \mathfrak{s}$ . Let $W$ be the set of these rational centres and define $\Sigma ^W=\bigcup _{w\in W}\Sigma _C^w$ . For any proper cluster $\mathfrak{s}\in \Sigma ^W$ fix a rational centre $w_{\mathfrak{s}}\in W$ . Denote $r_{\mathfrak{s}}=\frac{w_{\mathfrak{s}}-r}{\pi ^{\rho _{\mathfrak{s}}}}$ for $r\in \mathfrak{R}$ . Define reductions $\overline{f_{\mathfrak{s}}^W}(x)\in k[x]$ , $\overline{g_{\mathfrak{s}}}\in k[y]$ , and for $\mathfrak{s}\in \Sigma$ also $\overline{g_{\mathfrak{s}}^0}\in k[y]$ by

\begin{align*} \overline{f_{\mathfrak{s}}^W}(x^{b_{\mathfrak{s}}})&= \tfrac{u}{\pi ^{v(u)}}\prod _{r\in \mathfrak{s}\setminus \bigcup _{\mathfrak{s}^{\prime}\lt \mathfrak{s}}\mathfrak{s}^{\prime}} (x+r_{\mathfrak{s}})\mod \pi,& u&=c_f\textstyle \prod _{r\in \mathfrak{R}\setminus \mathfrak{s}} r_{\mathfrak{s}},\\ \overline{g_{\mathfrak{s}}}(y) &= y^{p_{\mathfrak{s}}/\gamma _{\mathfrak{s}}} - \tfrac{u}{\pi ^{v(u)}}\mod \pi, & u&=c_f\textstyle \prod _{r\in \mathfrak{R}\setminus \mathfrak{s}} r_{\mathfrak{s}},\\ \overline{g_{\mathfrak{s}}^0}(y) &= y^{p_{\mathfrak{s}}^0/\gamma _{\mathfrak{s}}^0} - \tfrac{u}{\pi ^{v(u)}}\mod \pi, & u&=c_f\textstyle \prod _{r\in \mathfrak{R}\setminus \{w_{\mathfrak{s}}\}} r_{\mathfrak{s}}. \end{align*}

where the union runs through all $\mathfrak{s}^{\prime}\in \Sigma ^W$ , $\mathfrak{s}^{\prime}\lt \mathfrak{s}$ . Finally define the $k$ -schemes

1. (1) $ X_{\mathfrak{s}}^W\;:\;\{{\overline{f_{\mathfrak{s}}^W}}=0\}\subset{\mathbb{G}}_{m,k}$ ;

2. (2) $ X_{\mathfrak{s}}\;:\;\{{\overline{g_{\mathfrak{s}}}}=0\}\subset{\mathbb{G}}_{m,k}$ ;

3. (3) $ X_{\mathfrak{s}}^0\;:\;\{{\overline{g_{\mathfrak{s}}^0}}=0\}\subset{\mathbb{G}}_{m,k}$ if $\mathfrak{s}\in \Sigma$ .

Notation 4.15. Given a scheme $\mathcal{X}/O_K$ we will denote by $\mathcal{X}_\eta$ its generic fibre $\mathcal{X}\times _{\textrm{Spec}\, O_K}\textrm{Spec}\, K$ , and by $\mathcal{X}_s$ its special fibre $\mathcal{X}\times _{\textrm{Spec}\, O_K}\textrm{Spec}\, k$ .

Notation 4.16. If $C=C_1\cup \dots \cup C_r$ is a chain of ${\mathbb{P}}^1_k$ s of length $r$ and multiplicities $m_i\in \mathbb{Z}$ (meeting transversely), then $\infty \in C_i$ is identified with $0\in C_{i+1}$ , and $0,\infty \in C$ are respectively $0\in C_{1}$ and $\infty \in C_{r}$ . Finally, if $r=0$ , then $C=\textrm{Spec}\, k$ and $0=\infty$ .

Notation 4.17. Let $\alpha \in \mathbb{Z}_+$ , $a,b\in \mathbb{Q}$ , with $a\gt b$ , and fix $\frac{n_i}{d_i}\in \mathbb{Q}$ so that

\begin{align*} \alpha a=\frac{n_0}{d_0}\gt \frac{n_1}{d_1}\gt \ldots \gt \frac{n_r}{d_r}\gt \frac{n_{r+1}}{d_{r+1}}=\alpha b,\quad \text{with} \quad \begin{vmatrix}n_i\!\!\!&n_{i+1}\\ d_i\!\!\!&d_{i+1}\\\end{vmatrix}=1, \end{align*}

and $r$ minimal. We write ${\mathbb{P}}^1(\alpha,a,b)$ for a chain of ${\mathbb{P}}^1_k$ s of length $r$ and multiplicities $\alpha d_1,\dots,\alpha d_r$ . Furthermore, we denote by ${\mathbb{P}}^1(\alpha,a)$ the chain ${\mathbb{P}}^1(\alpha,a,\lfloor \alpha a-1\rfloor/\alpha )$ .

Theorem 4.18 and Theorem 4.23 will follow from Section 5.

Theorem 4.18. Let $C/K$ be a hyperelliptic curve given by a Weierstrass equation $y^2=f(x)$ . Suppose $\deg f\gt 1$ and let $\Sigma$ , $W$ and $\Sigma ^W$ as in Definition 4.14 . Then there exists a proper flat model $\mathcal{C}/O_K$ (constructed in Section 5 ) of $C$ such that its special fibre $\mathcal{C}_s/k$ consists of $1$ -dimensional schemes given below in (1), (2), (3), (4), (5), glued along $0$ -dimensional transversal intersections:

1. (1) Every proper cluster $\mathfrak{s}\in \Sigma ^W$ gives a $1$ -dimensional closed subscheme $\Gamma _{\mathfrak{s}}$ of multiplicity $m_{\mathfrak{s}}$ . $\Gamma _{\mathfrak{s}}$ is not integral if and only if $D_{\mathfrak{s}}=2$ , $\tilde{\mathfrak{s}}\cap (\Sigma ^W\cup \{\varnothing \})=\varnothing$ and $\overline{f_{\mathfrak{s}}^W}$ is a square. When this happens, if $p=2$ then $\Gamma _{\mathfrak{s}}$ is not reduced and $(\Gamma _{\mathfrak{s}})_{\textrm{red}}$ is irreducible of multiplicity $2$ in $\Gamma _{\mathfrak{s}}$ , if $p\neq 2$ then $\Gamma _{\mathfrak{s}}$ is reducible, namely $\Gamma _{\mathfrak{s}}=\Gamma _{\mathfrak{s}}^+\cup \Gamma _{\mathfrak{s}}^-$ , with $\Gamma _{\mathfrak{s}}^{\pm }={\mathbb{P}}^1_k$ .

2. (2) Every proper cluster $\mathfrak{s}\in \Sigma ^W$ with $D_{\mathfrak{s}}=1$ gives the closed subscheme $X_{\mathfrak{s}}^W\times{\mathbb{P}}^1_k$ , of multiplicity $b_{\mathfrak{s}}$ , where $X_{\mathfrak{s}}^W\times \{0\}\subset \Gamma _{\mathfrak{s}}$ .

3. (3) Every proper cluster $\mathfrak{s}\in \Sigma ^W$ such that $\mathfrak{s}\neq \mathfrak{R}$ , gives the closed subscheme $X_{\mathfrak{s}}\times{\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}},s_{\mathfrak{s}}-p_{\mathfrak{s}}\cdot \frac{\rho _{\mathfrak{s}}-\rho _{P(\mathfrak{s})}}{2})$ where $X_{\mathfrak{s}}\times \{0\}\subset \Gamma _{\mathfrak{s}}$ and $X_{\mathfrak{s}}\times \{\infty \}\subset \Gamma _{P(\mathfrak{s})}$ .

4. (4) Every cluster $\mathfrak{s}\in \Sigma$ gives the closed subscheme $X_{\mathfrak{s}}^0\times{\mathbb{P}}^1(\gamma _{\mathfrak{s}}^0,-s_{\mathfrak{s}}^0)$ where $X_{\mathfrak{s}}^0\times \{0\}\subset \Gamma _{\mathfrak{s}}$ (the chains are open-ended).

5. (5) Finally, the cluster $\mathfrak{R}$ gives the closed subscheme $X_{\mathfrak{R}}\times{\mathbb{P}}^1(\gamma _{\mathfrak{R}},s_{\mathfrak{R}})$ where $X_{\mathfrak{R}}\times \{0\}\subset \Gamma _{\mathfrak{R}}$ (the chains are open-ended).

If $\Gamma _{\mathfrak{s}}$ is reducible, the two points in $X_{\mathfrak{s}}\times \{0\}$ (and $X_{\mathfrak{s}}^0\times \{0\}$ if $\mathfrak{s}\in \Sigma$ ) belong to different irreducible components of $\Gamma _{\mathfrak{s}}$ . Similarly, if $\mathfrak{s}\neq \mathfrak{R}$ and $\Gamma _{P(\mathfrak{s})}$ is reducible, the two points of $X_{\mathfrak{s}}\times \{\infty \}$ belong to different irreducible components of $\Gamma _{P(\mathfrak{s})}$ .

Furthermore, if $C$ has an almost rational cluster picture and is $y$ -regular, then, by choosing $\Sigma =\Sigma _C^{\textrm{min}}$ , the model $\mathcal{C}$ is regular with strict normal crossings. In that case, if $\mathfrak{s}$ is übereven and $\epsilon _{\mathfrak{s}}$ is even, then $\Gamma _{\mathfrak{s}}\simeq X_{\mathfrak{s}}\times{\mathbb{P}}^1_k$ , otherwise $\Gamma _{\mathfrak{s}}$ is irreducible of genus $g(\mathfrak{s})$ .

Remark 4.19. Consider the proper flat model $\mathcal{C}/O_K$ of Theorem 4.18. Via the canonical immersion $\mathcal{C}_s\hookrightarrow \mathcal{C}$ , the singular points of $\mathcal{C}$ are images of

• singular points of the subscheme given in (1) when $D_{\mathfrak{s}}=2$ and either $p=2$ , or $\mathfrak{s}=\mathfrak{t}^{\textrm{rat}}$ for some $\mathfrak{t}\in \Sigma _C$ with $|\mathfrak{t}|\gt |\rho _{\mathfrak{t}}|_p$ and $d_{\mathfrak{t}}\gt \rho _{\mathfrak{t}}$ , or $\mathfrak{s}={\mathfrak{s}}_1\wedge{\mathfrak{s}}_2$ for some ${\mathfrak{s}}_1\in \Sigma$ and ${\mathfrak{s}}_2\in \Sigma _C^{\textrm{min}}\smallsetminus \Sigma$ ;

• non-reduced points of the subscheme given in (2) when $D_{\mathfrak{s}}=1$ and either $\mathfrak{s}=\mathfrak{t}^{\textrm{rat}}$ for some $\mathfrak{t}\in \Sigma _C$ with $|\mathfrak{t}|\gt |\rho _{\mathfrak{t}}|_p$ and $d_{\mathfrak{t}}\gt \rho _{\mathfrak{t}}$ , or $\mathfrak{s}={\mathfrak{s}}_1\wedge{\mathfrak{s}}_2$ for some ${\mathfrak{s}}_1\in \Sigma$ and ${\mathfrak{s}}_2\in \Sigma _C^{\textrm{min}}\smallsetminus \Sigma$ ;

• non-reduced points of subschemes given in (3), (4), (5) (that may exist only if $p=2$ ).

Note that $\mathcal{C}$ is not necessarily normal, hence it may have infinitely many singular points.

Definition 4.20. Let $\mathfrak{s}\in \Sigma _{K^{nr}}$ . We say that

• $\mathfrak{s}$ is removable if either $|\mathfrak{s}|=1$ , or $\mathfrak{s}$ has a child $\mathfrak{s}^{\prime}\in \Sigma _{K^{nr}}$ of size $2g+1$ ( $\mathfrak{s}=\mathfrak{R}$ in this case).

• $\mathfrak{s}$ is contractible if

  1. (1) $|\mathfrak{s}|=2$ and $\rho _{\mathfrak{s}}\notin \mathbb{Z}$ , $\epsilon _{\mathfrak{s}}$ odd, $\rho _{P(\mathfrak{s})}\leq \rho _{\mathfrak{s}}-\frac{1}{2};$ or

  2. (2) $\mathfrak{s}=\mathfrak{R}$ of size $2g+2$ , with a child $\mathfrak{s}^{\prime}\in \Sigma _{K^{nr}}$ of size $2g$ , and $\rho _{\mathfrak{s}}\notin \mathbb{Z}$ , $v(c_f)$ odd, $\rho _{\mathfrak{s}^{\prime}}\geq \rho _{\mathfrak{s}}+\frac{1}{2}$ ; or

  3. (3) $\mathfrak{s}=\mathfrak{R}$ of size $2g+2$ , union of its $2$ odd proper children ${\mathfrak{s}}_1,{\mathfrak{s}}_2\in \Sigma _{K^{nr}}$ , with $v(c_f)$ odd, $\rho _{{\mathfrak{s}}_i}\geq \rho _{\mathfrak{s}}+1$ for $i=1,2$ .

Notation 4.21. Write $\Sigma ^{nr}\subseteq \Sigma _{K^{nr}}$ for the subset of non-removable clusters.

Definition 4.22. Choose rational centres $w_{\mathfrak{s}}\in K^{nr}$ for every $\mathfrak{s}\in \Sigma ^{nr}$ , in such a way that $w_{\mathfrak{s}}\in \mathfrak{s}$ when $p_{\mathfrak{s}}^0=1$ , and $\sigma (w_{\mathfrak{s}})=w_{\sigma (\mathfrak{s})}$ for all $\sigma \in \textrm{Gal}(K^{nr}/K)$ . Denote $r_{\mathfrak{s}}=\frac{w_{\mathfrak{s}}-r}{\pi ^{\rho _{\mathfrak{s}}}}$ for $r\in \mathfrak{R}$ and define $\overline{g_{\mathfrak{s}}}, \overline{g_{\mathfrak{s}}^0}\in k^{\textrm{s}}[y]$ as in Definition 4.14, and $\overline{f_{\mathfrak{s}}}(x)\in k^{\textrm{s}}[x]$ , by

\begin{align*} x^{2-p_{\mathfrak{s}}^0}\overline{f_{\mathfrak{s}}}(x^{b_{\mathfrak{s}}}) = \tfrac{u}{\pi ^{v(u)}} \prod _{r\in \mathfrak{s}\setminus \bigcup _{\mathfrak{s}^{\prime}\lt \mathfrak{s}}\mathfrak{s}^{\prime}} (x+r_{\mathfrak{s}})\mod \pi,\quad u=c_f\textstyle \prod _{r\in \mathfrak{R}\setminus \mathfrak{s}} r_{\mathfrak{s}}, \end{align*}

where the union runs through all $\mathfrak{s}^{\prime}\in \Sigma ^{nr}$ , $\mathfrak{s}^{\prime}\lt \mathfrak{s}$ . Let $G_{\mathfrak{s}}=\textrm{Stab}_{G_K}(\mathfrak{s})$ , $K_{\mathfrak{s}}=\left ( K^{\textrm{s}}\right )^{G_{\mathfrak{s}}}$ , and let $k_{\mathfrak{s}}$ be the residue field of $K_{\mathfrak{s}}$ . Then ${\overline{f_{\mathfrak{s}}}}\in k_{\mathfrak{s}}[x]$ , ${\overline{g_{\mathfrak{s}}}}\in k_{\mathfrak{s}}[y]$ , and for $\mathfrak{s}$ minimal ${\overline{g_{\mathfrak{s}}^0}}\in k_{\mathfrak{s}}[y]$ .

Let ${\mathfrak{s}}_0\in \Sigma ^{nr}$ be minimal and contained in $\mathfrak{s}$ . Denote $\mathfrak{s}=\tilde{\mathfrak{s}}\setminus \{\{r\}\lt \mathfrak{s}\mid r\neq w_{{\mathfrak{s}}_0}\}$ . Note that $\mathfrak{s}$ does not depend on the choice of ${\mathfrak{s}}_0$ . Define $\tilde{f_{\mathfrak{s}}}\in k_{\mathfrak{s}}[x]$ by

\begin{align*} \tilde{f}_{\mathfrak{s}}(x)=\prod _{\mathfrak{s}^{\prime}\in \mathfrak{s}}\left ( x-{\overline{u_{\mathfrak{s}^{\prime},\mathfrak{s}}}}\right )\cdot \overline{f_{\mathfrak{s}}}(x), \end{align*}

where ${\overline{u_{\mathfrak{s}^{\prime},\mathfrak{s}}}}=\frac{w_{\mathfrak{s}^{\prime}}-w_{\mathfrak{s}}}{\pi ^{\rho _{\mathfrak{s}}}}\mod \pi$ if $\mathfrak{s}^{\prime}\neq \varnothing$ and ${\overline{u_{\mathfrak{s}^{\prime},\mathfrak{s}}}}=0$ otherwise.

In the next theorem we describe the special fibre of the minimal regular model of $C$ with normal crossings. We use Definitions/Notations 3.1, 3.3, 3.4, 3.2, 3.8, 3.9, 3.26, 4.6, 4.10, 4.13, 4.17, 4.20, 4.21, 4.22 in the statement. Note that a full description of the model is developed in Section 5.

Theorem 4.23 (Minimal regular NC model). Let $C/K\; : \;y^2=f(x)$ be a hyperelliptic curve of genus $\geq 1$ . Suppose $C_{K^{nr}}$ has an almost rational cluster picture and is $y$ -regular. Then the minimal regular model with normal crossings $\mathcal{C}^{\textrm{min}}/O_{K^{nr}}$ of $C$ has special fibre $\mathcal{C}_s^{\textrm{min}}/k^{\textrm{s}}$ described as follows:

1. (1) Every $\mathfrak{s}\in \Sigma ^{nr}$ gives a $1$ -dimensional subscheme $\Gamma _{\mathfrak{s}}$ of multiplicity $m_{\mathfrak{s}}$ . If $\mathfrak{s}$ is übereven and $\epsilon _{\mathfrak{s}}$ is even, then $\Gamma _{\mathfrak{s}}$ is the disjoint union of $\Gamma _{\mathfrak{s}}^{r_{\mathfrak{s},-}}\simeq{\mathbb{P}}^1$ and $\Gamma _{\mathfrak{s}}^{r_{\mathfrak{s},+}}\simeq{\mathbb{P}}^1$ , otherwise $\Gamma _{\mathfrak{s}}$ is irreducible of genus $g(\mathfrak{s})$ (write $\Gamma _{\mathfrak{s}}^{r_{\mathfrak{s},-}}=\Gamma _{\mathfrak{s}}^{r_{\mathfrak{s},+}}=\Gamma _{\mathfrak{s}}$ in this case). The indices $r_{\mathfrak{s},-}$ and $r_{\mathfrak{s},+}$ are the roots of $\overline{g_{\mathfrak{s}}}$ (where $r_{\mathfrak{s},-}=r_{\mathfrak{s},+}$ if $\deg{\overline{g_{\mathfrak{s}}}}=1$ ).

2. (2) Every $\mathfrak{s}\in \Sigma ^{nr}$ with $D_{\mathfrak{s}}=1$ gives open-ended ${\mathbb{P}}^1$ s of multiplicity $b_{\mathfrak{s}}$ from $\Gamma _{\mathfrak{s}}$ indexed by roots of $\overline{f_{\mathfrak{s}}}$ .

3. (3) Every non-maximal element $\mathfrak{s}\in \Sigma ^{nr}$ gives chains ${\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}},s_{\mathfrak{s}}-p_{\mathfrak{s}}\cdot \frac{\rho _{\mathfrak{s}}-\rho _{P(\mathfrak{s})}}{2})$ from $\Gamma _{\mathfrak{s}}$ to $\Gamma _{P(\mathfrak{s})}$ indexed by roots of $\overline{g_{\mathfrak{s}}}$ .

4. (4) Every minimal element $\mathfrak{s}\in \Sigma ^{nr}$ gives open-ended chains ${\mathbb{P}}^1(\gamma _{\mathfrak{s}}^0,-s_{\mathfrak{s}}^0)$ from $\Gamma _{\mathfrak{s}}$ indexed by roots of $\overline{g_{\mathfrak{s}}^0}$ .

5. (5) The maximal element $\mathfrak{s}\in \Sigma ^{nr}$ gives open-ended chains ${\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}})$ from $\Gamma _{\mathfrak{s}}$ indexed by roots of $\overline{g_{\mathfrak{s}}}$ .

6. (6) Finally, blow down all $\Gamma _{\mathfrak{s}}$ where $\mathfrak{s}$ is a contractible cluster.

In (3) and (5), a chain indexed by $r$ goes from $\Gamma _{\mathfrak{s}}^r$ . In (3) the chain indexed by $r_{\mathfrak{s},-}$ goes to $\Gamma _{P(\mathfrak{s})}^{r_{P(\mathfrak{s}),-}}$ and the chain indexed by $r_{\mathfrak{s},+}$ goes to $\Gamma _{P(\mathfrak{s})}^{r_{P(\mathfrak{s}),+}}$ .

Before blowing down in (6), the components given in (1)–(5) describe the special fibre of a regular model of $C_{K^{nr}}$ with strict normal crossings.

The Galois group $G_k$ acts naturally, that is for every $\sigma \in G_k$ , $\sigma (\Gamma _{\mathfrak{s}}^r)=\Gamma _{\sigma (\mathfrak{s})}^{\sigma (r)}$ , and similarly, on the chains.

If $\Gamma _{\mathfrak{s}}$ is irreducible, then its function field is isomorphic to $k^{\textrm{s}}(x)[y]$ with the relation $y^{D_{\mathfrak{s}}}=\tilde{f_{\mathfrak{s}}}(x)$ .

Remark 4.24. Note that if $\Gamma _{\mathfrak{s}}$ or $\Gamma _{P(\mathfrak{s})}$ is reducible then $p_{\mathfrak{s}}/\gamma _{\mathfrak{s}}=2$ .

Example 4.25. Let $p$ be a prime number and let $a\in \mathbb{Z}_p$ , $b\in \mathbb{Z}_p^\times$ such that the polynomial $x^2+ax+b$ is not a square modulo $p$ . Let $C$ be the hyperelliptic curve over $\mathbb{Q}_p$ of genus $4$ given by the equation $y^2=f(x)$ , where $f(x)=(x^6+ap^4x^3+bp^8)((x-p)^3-p^{11})$ . In Example 3.32 , we described the rational cluster picture of $C$ and proved that $C$ has an almost rational cluster picture. Recall that $\Sigma _C^{\textrm{rat}}$ consists of $3$ clusters $\mathfrak{t}_3,\mathfrak{t}_4,\mathfrak{R}$ of size $6,3,9$ respectively such that $\mathfrak{t}_3\lt \mathfrak{R}$ and $\mathfrak{t}_4\lt \mathfrak{R}$ . In particular, note that $\Sigma _{\mathbb{Q}_p^{nr}}=\Sigma _C^{\textrm{rat}}$ , and no cluster of $\Sigma _{\mathbb{Q}_p^{nr}}$ is removable, so $\Sigma ^{nr}=\Sigma _C^{\textrm{rat}}$ . The minimal elements of $\Sigma ^{nr}$ are then $\mathfrak{t}_3$ and $\mathfrak{t}_4$ .

We want to describe the special fibre of the minimal regular model with normal crossings $\mathcal{C}^{\textrm{min}}$ of $C$ . Compute the quantities in Definitions 4.6 and 4.13 , and the polynomials ${\overline{f_{\mathfrak{s}}}},{\overline{g_{\mathfrak{s}}}},{\overline{g_{\mathfrak{s}}^0}}$ of Definition 4.22 , for any cluster in $\Sigma ^{nr}$ :

	$\rho _{\mathfrak{s}}$	$b_{\mathfrak{s}}$	$\epsilon _{\mathfrak{s}}$	$D_{\mathfrak{s}}$	$m_{\mathfrak{s}}$	$p_{\mathfrak{s}}$	$s_{\mathfrak{s}}$	$\gamma _{\mathfrak{s}}$	$p_{\mathfrak{s}}^0$	$s_{\mathfrak{s}}^0$	$\gamma _{\mathfrak{s}}^0$	$g(\mathfrak{s})$	${\overline{f_{\mathfrak{s}}}}(x)$	${\overline{g_{\mathfrak{s}}}}(y)$	${\overline{g_{\mathfrak{s}}^0}}(y)$
$\mathfrak{t}_3$	$\tfrac{4}{3}$	$3$	$11$	$1$	$6$	$2$	$-\tfrac{1}{6}$	$2$	$2$	$-\tfrac{25}{6}$	$2$	$0$	$x^2+\bar{ax}+\bar{b}$	$y+1$	$y-1$
$\mathfrak{t}_4$	$\tfrac{11}{3}$	$3$	$17$	$1$	$6$	$1$	$-\tfrac{7}{6}$	$1$	$2$	$-\tfrac{29}{6}$	$2$	$0$	$x-1$	$y-1$	$y+1$
$\mathfrak{R}$	$1$	$1$	$9$	$1$	$2$	$1$	$\tfrac{1}{2}$	$1$	$2$			$0$	$1$	$y-1$

where $\bar{a}, \bar{b}$ are the reductions of $a,b$ modulo $p$ . Then $C$ is also $y$ -regular for any $p$ . Following the steps of Theorem 4.23 the special fibre of $\mathcal{C}^{\textrm{min}}$ over $\bar{\mathbb{F}}_p$ can be described as follows:

1. (1) The clusters $\mathfrak{t}_3,\mathfrak{t}_4,\mathfrak{R}$ give $3$ irreducible components $\Gamma _{\mathfrak{t}_3},\Gamma _{\mathfrak{t}_4},\Gamma _{\mathfrak{R}}$ of genus $0$ of multiplicities $6,6,2$ respectively;

2. (2) The cluster $\mathfrak{t}_3$ gives $2$ open-ended ${\mathbb{P}}^1$ s of multiplicity $3$ from $\Gamma _{\mathfrak{t}_3}$ , while $\mathfrak{t}_4$ gives $1$ open-ended ${\mathbb{P}}^1$ of multiplicity $3$ from $\Gamma _{\mathfrak{t}_4}$ .

3. (3) From $\gamma _{\mathfrak{t}_3}s_{\mathfrak{t}_3}=-\tfrac{1}{3}\gt -\tfrac{1}{2}\gt -1=\gamma _{\mathfrak{t}_3}\big (s_{\mathfrak{t}_3}-p_{\mathfrak{t}_3}\cdot \tfrac{\rho _{\mathfrak{t}_3}-\rho _{\mathfrak{R}}}{2}\big ),$ the cluster $\mathfrak{t}_3$ gives $1$ ${\mathbb{P}}^1$ of multiplicity $4$ from $\Gamma _{\mathfrak{t}_3}$ to $\Gamma _{\mathfrak{R}}$ . From

   \begin{align*} \gamma _{\mathfrak{t}_4}s_{\mathfrak{t}_4}=-\tfrac{7}{6}\gt -\tfrac{6}{5}\gt -\tfrac{5}{4}\gt -\tfrac{4}{3}\gt -\tfrac{3}{2}\gt -2\gt -\tfrac{5}{2}=\gamma _{\mathfrak{t}_3}\big (s_{\mathfrak{t}_4}-p_{\mathfrak{t}_4}\cdot \tfrac{\rho _{\mathfrak{t}_4}-\rho _{\mathfrak{R}}}{2}\big ) \end{align*}
   the cluster $\mathfrak{t}_4$ gives $1$ chain of ${\mathbb{P}}^1$ s of multiplicities $5,4,3,2,1$ from $\Gamma _{\mathfrak{t}_4}$ to $\Gamma _{\mathfrak{R}}$ .

4. (4) From $-\gamma _{\mathfrak{t}_3}^0s_{\mathfrak{t}_3}^0=\tfrac{25}{3}\gt 8\gt 7$ the cluster $\mathfrak{t}_3$ gives $1$ open-ended ${\mathbb{P}}^1$ of multiplicity $2$ from $\Gamma _{\mathfrak{t}_3}$ . From $-\gamma _{\mathfrak{t}_4}^0s_{\mathfrak{t}_4}^0=\tfrac{29}{3}\gt \tfrac{19}{2}\gt 9\gt 8$ , the cluster $\mathfrak{t}_4$ gives $1$ open-ended chain of ${\mathbb{P}}^1$ s of multiplicities $4,2$ from $\Gamma _{\mathfrak{t}_4}$ .

5. (5) From $\gamma _{\mathfrak{R}}s_{\mathfrak{R}}=\tfrac{1}{2}\gt 0\gt -1$ , the cluster $\mathfrak{R}$ gives $1$ open-ended ${\mathbb{P}}^1$ of multiplicity $1$ from $\Gamma _{\mathfrak{R}}$ .

6. (6) There is no contractible cluster, so the components we considered in the steps above describe the special fibre of $\mathcal{C}^{\textrm{min}}$ over $\bar{\mathbb{F}}_p$ :

   

Finally, from the Galois action on the roots of the polynomials ${\overline{f_{\mathfrak{s}}}},{\overline{g_{\mathfrak{s}}}},{\overline{g_{\mathfrak{s}}^0}}$ , for $\mathfrak{s}\in \Sigma ^{nr}$ , we get that $G_k$ acts trivially if $x^2+\bar{a} x+\bar{b}$ is reducible in ${\mathbb{F}}_p$ , while it swaps the two components of multiplicity $3$ intersecting $\Gamma _{\mathfrak{t}_3}$ (coming from (2)) otherwise.

As application of Theorem 4.23 we suppose $k$ is finite of characteristic $p\gt 2$ and $C$ is semistable of genus $g\geq 2$ . In this setting [14, Theorem 8.5] describes the minimal regular model of $C$ in terms of its cluster picture $\Sigma _C$ . We compare that result with the one obtained from Theorem 4.23 (Corollary 4.27).

First note that $C_{K^{nr}}$ is $y$ -regular as $p\neq 2$ . From [14, Definition 1.7], if $C$ is semistable then

1. (1) the extension $K(\mathfrak{R})/K$ has ramification degree at most $2$ ;

2. (2) every proper cluster is $\textrm{Gal}(K^{\textrm{s}}/K^{nr})$ -invariant;

3. (3) every principal cluster has $d_{\mathfrak{s}}\in \mathbb{Z}$ and $\nu _{\mathfrak{s}}\in 2\mathbb{Z}$ .

It follows from Corollary 3.27 that $C_{K^{nr}}$ has an almost rational cluster picture.

In fact, (1) and (2) imply $\rho _{\mathfrak{s}}=d_{\mathfrak{s}}$ and $\epsilon _{\mathfrak{s}}=\nu _{\mathfrak{s}}$ for any proper cluster $\mathfrak{s}$ (Remark 3.13). In particular, $\Sigma _{C_{K^{nr}}}^{\textrm{rat}}=\Sigma _C$ . We will then say that $\mathfrak{s}\in \Sigma _C$ is non-removable if $\mathfrak{s}$ is proper and non-removable as rational cluster in $\Sigma _{K^{nr}}$ .

Lemma 4.26. Suppose $k$ finite and $p\gt 2$ . Assume $C$ is semistable and let $\mathfrak{s}\in \Sigma _C$ be a non-removable cluster. Then $d_{\mathfrak{s}}\in \frac{1}{2}\mathbb{Z}$ and $\nu _{\mathfrak{s}}\in \mathbb{Z}$ . Moreover, $\mathfrak{s}$ is contractible if and only if $d_{\mathfrak{s}}\notin \mathbb{Z}$ or $\nu _{\mathfrak{s}}\notin 2\mathbb{Z}$ .

Proof. Let $\mathfrak{s}\in \Sigma _C$ be a non-removable cluster. Since $K(\mathfrak{R})/K$ has ramification degree at most $2$ , then $d_{\mathfrak{s}}\in \tfrac{1}{2}\mathbb{Z}$ .

By Theorem 4.23 the multiplicity of the $1$ -dimensional scheme $\Gamma _{\mathfrak{s}}$ is $m_{\mathfrak{s}}$ . Furthermore, $\Gamma _{\mathfrak{s}}$ is an irreducible component of the special fibre of the minimal regular model of $C$ if and only if $\mathfrak{s}$ is not contractible. Therefore if $\mathfrak{s}$ is not contractible, then $m_{\mathfrak{s}}=1$ , that is $D_{\mathfrak{s}}=2$ and $b_{\mathfrak{s}}=1$ . It follows that $\nu _{\mathfrak{s}}\in 2\mathbb{Z}$ and $d_{\mathfrak{s}}\in \mathbb{Z}$ . Suppose $\mathfrak{s}$ contractible. Then either $d_{\mathfrak{s}}\notin \mathbb{Z}$ (and $\nu _{\mathfrak{s}}\in \mathbb{Z}$ ) or $\mathfrak{s}=\mathfrak{R}$ of size $2g+2$ , with $2$ odd rational children and $v(c_f)$ odd. We want to show that in the latter case, $\nu _{\mathfrak{s}}$ is odd. By Lemma 3.18, $d_{\mathfrak{R}}\in \mathbb{Z}$ . Then $\nu _{\mathfrak{R}}=v(c_f)+(2g+2)d_{\mathfrak{R}}$ is odd.

Let $\mathfrak{s}\in \Sigma _C$ be a non-removable cluster. By Lemma 4.26, if $\mathfrak{s}$ is not contractible, then $2g(\mathfrak{s})+1$ or $2g(\mathfrak{s})+2$ equals the number of odd children of $\mathfrak{s}$ . In fact, this also holds when $\mathfrak{s}$ is contractible since in that case $g(\mathfrak{s})=0$ and $\mathfrak{s}$ has at most $2$ odd children.

Corollary 4.27 (Minimal regular model (semistable reduction)). Suppose that $k$ is finite and $p\gt 2$ . Let $C/K$ be a semistable hyperelliptic curve of genus $g\geq 2$ . The minimal regular model $\mathcal{C}^{\textrm{min}}/O_{K^{nr}}$ of $C$ has special fibre $\mathcal{C}_s^{\textrm{min}}/k^{\textrm{s}}$ described as follows:

1. (1) Every non-removable cluster $\mathfrak{s}\in \Sigma _C$ gives a $1$ -dimensional subscheme $\Gamma _{\mathfrak{s}}$ . If $\mathfrak{s}$ is übereven, then $\Gamma _{\mathfrak{s}}$ is the disjoint union of $\Gamma _{\mathfrak{s}}^{r_{\mathfrak{s},-}}\simeq{\mathbb{P}}^1$ and $\Gamma _{\mathfrak{s}}^{r_{\mathfrak{s},+}}\simeq{\mathbb{P}}^1$ , otherwise $\Gamma _{\mathfrak{s}}$ is irreducible of genus $g(\mathfrak{s})$ (write $\Gamma _{\mathfrak{s}}^{r_{\mathfrak{s},-}}=\Gamma _{\mathfrak{s}}^{r_{\mathfrak{s},+}}=\Gamma _{\mathfrak{s}}$ in this case). The indices $r_{\mathfrak{s},-}$ and $r_{\mathfrak{s},+}$ are the roots of $\overline{g_{\mathfrak{s}}}$ .

2. (2) Every odd proper cluster $\mathfrak{s}\in \Sigma _C$ , with size $|\mathfrak{s}|\leq 2g$ , gives a chain of ${\mathbb{P}}^1$ s of length $\big \lfloor \frac{d_{\mathfrak{s}}-d_{P(\mathfrak{s})}-1}{2}\big \rfloor$ from $\Gamma _{\mathfrak{s}}$ to $\Gamma _{P(\mathfrak{s})}$ indexed by the root of $\overline{g_{\mathfrak{s}}}$ .

3. (3) Every even proper cluster $\mathfrak{s}\in \Sigma _C$ , with size $|\mathfrak{s}|\leq 2g$ , gives a chain of ${\mathbb{P}}^1$ s of length $\left \lfloor d_{\mathfrak{s}}-d_{P(\mathfrak{s})}-\frac{1}{2}\right \rfloor$ from $\Gamma _{\mathfrak{s}}^{r_{\mathfrak{s},-}}$ to $\Gamma _{P(\mathfrak{s})}^{r_{P(\mathfrak{s}),-}}$ indexed by $r_{\mathfrak{s},-}$ and a chain of ${\mathbb{P}}^1$ s of same length from $\Gamma _{\mathfrak{s}}^{r_{\mathfrak{s},+}}$ to $\Gamma _{P(\mathfrak{s})}^{r_{P(\mathfrak{s}),+}}$ indexed by $r_{\mathfrak{s},+}$ .

4. (4) Finally, blow down all $\Gamma _{\mathfrak{s}}$ where $\mathfrak{s}$ is a contractible cluster.

All components have multiplicity $1$ , and the absolute Galois group $G_k$ acts naturally, as in Theorem 4.23.

Proof. Let $\mathfrak{s}\in \Sigma _C$ be a non-removable cluster. From Lemma 4.26, if $\mathfrak{s}$ is not contractible, then $D_{\mathfrak{s}}=2$ , $\gamma _{\mathfrak{s}} s_{\mathfrak{s}}\in \mathbb{Z}$ and $\gamma _{\mathfrak{s}}^0 s_{\mathfrak{s}}^0\in \mathbb{Z}$ . Suppose $\mathfrak{s}$ contractible. If $|\mathfrak{s}|=2$ with $d_{\mathfrak{s}}\notin \mathbb{Z}$ (case (1) of Definition 4.20), then $\gamma _{\mathfrak{s}}^0 s_{\mathfrak{s}}^0\in \mathbb{Z}$ and $\gamma _{\mathfrak{s}}=1$ , $s_{\mathfrak{s}}\in \tfrac{1}{2}\mathbb{Z}\setminus \mathbb{Z}$ and so $s_{\mathfrak{s}}-d_{\mathfrak{s}}+d_{P(\mathfrak{s})}\in \mathbb{Z}$ , as $P(\mathfrak{s})$ can not be contractible. If $\mathfrak{s}=\mathfrak{R}$ (cases (2), (3) of Definition 4.20), then $v(c_f)$ is odd, and so $\gamma _{\mathfrak{s}}=2$ and $\gamma _{\mathfrak{s}} s_{\mathfrak{s}}\in \mathbb{Z}$ . Therefore (2), (4) and (5) of Theorem 4.23 do not give any components.

Finally, as $\gamma _{\mathfrak{s}}=1$ and $p_{\mathfrak{s}}\frac{d_{\mathfrak{s}}-d_{P(\mathfrak{s})}}{2}\in \frac{1}{2}\mathbb{Z}$ for any proper $\mathfrak{s}$ with size $|\mathfrak{s}|\leq 2g$ (i.e. non-maximal), the length of ${\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}},s_{\mathfrak{s}}-p_{\mathfrak{s}}\cdot \frac{d_{\mathfrak{s}}-d_{P(\mathfrak{s})}}{2})$ is

\begin{align*} \left \lfloor \gamma _{\mathfrak{s}} s_{\mathfrak{s}}-\gamma _{\mathfrak{s}}\left ( s_{\mathfrak{s}}-p_{\mathfrak{s}}\cdot \frac{d_{\mathfrak{s}}-d_{P(\mathfrak{s})}}{2}\right )-\frac{1}{2}\right \rfloor =\left \lfloor p_{\mathfrak{s}}\cdot \frac{d_{\mathfrak{s}}-d_{P(\mathfrak{s})}}{2}-\frac{1}{2}\right \rfloor. \end{align*}

The Corollary then follows from Theorem 4.23.

5. Construction of the model

We are going to construct a proper flat model $\mathcal{C}/O_K$ of $C$ by glueing models defined in [1, §4]. For this reason we will assume the reader has familiarity with the definitions and the results presented in that paper. Let us start this section by describing the strategy we will follow.

Let $\Sigma _C^{\textrm{min}}$ be the set of rationally minimal clusters of $C$ and let $\Sigma \subseteq \Sigma _C^{\textrm{min}}$ non-empty. For any cluster $\mathfrak{s}\in \Sigma$ fix a rational centre $w_{\mathfrak{s}}$ in such a way that $\mathring{\Sigma }_C^{w_{\mathfrak{s}}}$ consists of the proper clusters in $\Sigma _C^{w_{\mathfrak{s}}}$ . This requirement can be satisfied by choosing $w_{\mathfrak{s}}\in \mathfrak{s}$ whenever possible.3 Let $W$ be the set of all such rational centres and define $\Sigma ^W\;:\!=\;\bigcup _{w\in W}\Sigma _C^{w}$ . For every proper cluster $\mathfrak{t}\in \Sigma ^W$ fix a rational centre $w_{\mathfrak{t}}\in W$ (Lemma 3.14). For every $w\in W$ , consider the curve $C^{w}\;:\;y^2=f(x+w)$ , isomorphic to $C$ , and construct the (proper flat) model $\mathcal{C}_\Delta ^{w}/O_K$ by [1, §4, Theorem 3.14]. We will define an open subscheme $\mathring{\mathcal{C}}_\Delta ^{w}$ of $\mathcal{C}_\Delta ^{w}$ and we will show that glueing the schemes $\mathring{\mathcal{C}}_\Delta ^{w}$ , to varying of $w\in W$ , along common opens, gives a proper flat model $\mathcal{C}/O_K$ of $C$ . Furthermore, if $\Sigma =\Sigma _C^{\textrm{min}}$ , and $C$ is $y$ -regular and has an almost rational cluster picture, then $\mathring{\mathcal{C}}_\Delta ^{w}$ is an open regular subscheme of $\mathcal{C}_\Delta ^{w}$ and therefore $\mathcal{C}$ is also regular.

5.1. Charts

Let $\Sigma =\{{\mathfrak{s}}_1\dots,{\mathfrak{s}}_m\}\subseteq \Sigma _C^{\textrm{min}}$ be a non-empty set of rationally minimal clusters and let $W=\{w_1,\dots,w_m\}$ be a set of corresponding rational centres, such that $\mathring{\Sigma }_C^{w_h}$ consists of the proper clusters of $\Sigma _C^{w_h}$ , for any $h=1,\dots,m$ . Define $\Sigma ^W\;:\!=\;\bigcup _{h=1}^m\Sigma _C^{w_h}$ . For any $h,l=1,\dots,m$ , $h\neq l$ , define $w_{hl}\;:\!=\;w_h-w_l$ , and write $w_{hl}=u_{hl}\pi ^{\rho _{hl}}$ , where $u_{hl}\in O_K^\times$ and $\rho _{hl}\in \mathbb{Z}$ . Note that $\rho _{hl}=\rho _{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}=\rho _{lh}$ , by Lemma 3.18. Set $u_{hh}\;:\!=\;0$ . Finally, for any $h,l=1,\dots,m$ , denote by $\overline{u_{hl}}\in k$ the reduction of $u_{hl}$ modulo $\pi$ .

Definition 5.1. Let $h=1,\dots,m$ and let $\mathfrak{t}\in \Sigma _C^{w_h}$ be a proper cluster. Recall the matrices and cones defined in [ 1, §4]. We say that a matrix $M$ is associated to $\mathfrak{t}$ if either

1. (i) $M=M_{L,i}$ , with $L=L_{\mathfrak{t}}^{w_h}$ and $i=0,\dots,r_L$ or

2. (ii) $M=M_{V,j}$ , with $V=V_{\mathfrak{t}}^{w_h}$ and $i=0,\dots,r_V$ or

3. (iii) $M=M_{V_0,j}$ , with $V_0=V_0^{w_h}$ and $j=0,\dots,r_{V_0}$ , if $\mathfrak{t}={\mathfrak{s}}_h$ .

For a matrix $M$ associated to $\mathfrak{t}$ we denote by $\delta _M$ and $\sigma _M$ respectively

1. (i) the denominator $\delta _{L_{\mathfrak{t}}^{w_h}}$ and the cone $\sigma _{L_{\mathfrak{t}}^{w_h},i,i+1}$ if $M=M_{L_{\mathfrak{t}}^{w_h},i}$ ,

2. (ii) the denominator $\delta _{V_{\mathfrak{t}}^{w_h}}$ and the cone $\sigma _{V_{\mathfrak{t}}^{w_h},j,j+1}$ if $M=M_{V_{\mathfrak{t}}^{w_h},j}$ ,

3. (iii) the denominator $\delta _{V_0^{w_h}}$ and the cone $\sigma _{V_0^{w_h},j,j+1}$ if $M=M_{V_0^{w_h},j}$ .

Finally, define $X_M=\textrm{Spec}\, O_K[\sigma _M^\vee \cap \mathbb{Z}^3]$ and write

\begin{align*} X_\Delta ^h=\bigcup X_M, \end{align*}

for the toric scheme defined in [ 1, §4.2] from the Newton polytope $\Delta _v^{w_h}$ associated to the curve $C^{w_h}$ . Therefore, by Lemma 4.3 , the union runs through every proper cluster $\mathfrak{t}\in \Sigma _C^{w_h}$ and all matrices $M$ associated to $\mathfrak{t}$ .

The following Lemma describes all possible matrices associated to $\mathfrak{t}$ .

Lemma 5.2. Let $\mathfrak{t}\in \Sigma _C^{w_h}$ be a proper cluster. Consider the $v$ -face $F_{\mathfrak{t}}^{w_h}$ . Let $P_0,P_1\in \mathbb{Z}^2$ and $n_i,d_i,k_i\in \mathbb{Z}$ be as in [ 1, §4] and define

\begin{align*} \delta \;:\!=\;\delta _M,\quad \gamma _i\;:\!=\;\frac{n_0d_i-n_id_0}{\delta d_0}\quad \mbox{ and }\quad T_i\;:\!=\;\left (\begin{matrix} \frac{1}{\delta } && -k_i&& k_{i+1}\\ 0 && \delta && 0\\ 0 && 0 && \delta \end{matrix}\right ), \end{align*}

for each matrix $M$ associated to $\mathfrak{t}$ .

• Let $c$ be the unique element of $\{0,\dots,b_{\mathfrak{t}}-1\}$ such that $\tfrac{1}{b_{\mathfrak{t}}}-\rho _{\mathfrak{t}}\cdot c=d\in \mathbb{Z}$ . For $L=L_{\mathfrak{t}}^{w_h}$ and for all $i=0,\dots,r_{L}$ , choose $k_i=cn_i+d\delta d_i(\lfloor \mathfrak{t}/2\rfloor +1)$ ; then $M_{L,i}$ and $M_{L,i}^{-1}$ are respectively

  \begin{align*} \left (\begin{matrix} \delta && -c\delta d_i\left (\tfrac{\epsilon _{\mathfrak{t}}}{2}+\gamma _i\right ) && c\delta d_{i+1}\left (\tfrac{\epsilon _{\mathfrak{t}}}{2}+\gamma _{i+1}\right )\\[4pt] 0 && d_i && -d_{i+1}\\[4pt] -\delta \rho _{\mathfrak{t}} && -d\delta d_i\left (\tfrac{\epsilon _{\mathfrak{t}}}{2}+\gamma _i\right ) && d\delta d_{i+1}\left (\tfrac{\epsilon _{\mathfrak{t}}}{2}+\gamma _{i+1}\right ) \end{matrix}\right ),\qquad T_i \cdot \left (\begin{matrix} 1 && \left \lfloor \frac{|\mathfrak{t}|}{2}\right \rfloor +1 && 0\\[4pt] d_{i+1}\rho _{\mathfrak{t}} && \frac{d_{i+1}\epsilon _{\mathfrak{t}}}{2}+\gamma _{i+1} && d_{i+1}\\[4pt] d_i\rho _{\mathfrak{t}} && \frac{d_i\epsilon _{\mathfrak{t}}}{2}+\gamma _i && d_i \end{matrix}\right ), \end{align*}
  where $P_0=(|\mathfrak{t}|,0)$ , $P_1=(\lfloor{|\mathfrak{t}|-1}/{2}\rfloor,1)$ and $\delta =\delta _{L}=b_{\mathfrak{t}}$ .

• If $\mathfrak{t}$ is odd, then for $V=V_{\mathfrak{t}}^{w_h}$ and for all $j=0,\dots, r_{V}$ , the matrices $M_{V,j}$ and $M_{V,j}^{-1}$ are respectively

  \begin{align*} \left (\begin{matrix} -|\mathfrak{t}| && -\frac{|\mathfrak{t}|+1}{2} d_j && \frac{|\mathfrak{t}|+1}{2}d_{j+1}\\[4pt] 2 && d_j &&-d_{j+1}\\[4pt] -\epsilon _{\mathfrak{t}}+|\mathfrak{t}|\rho _{\mathfrak{t}} && n_j &&-n_{j+1} \end{matrix}\right ),\qquad T_j\cdot \left (\begin{matrix} 1 && \frac{|\mathfrak{t}|+1}{2} && 0\\[4pt] d_{j+1}\rho _{\mathfrak{t}} -2\cdot \gamma _{j+1} && \frac{d_{j+1}\epsilon _{\mathfrak{t}}}{2}-|\mathfrak{t}|\cdot \gamma _{j+1}&& d_{j+1}\\[4pt] d_j\rho _{\mathfrak{t}} -2\cdot \gamma _{j} &&\frac{d_j\epsilon _{\mathfrak{t}}}{2}-|\mathfrak{t}|\cdot \gamma _j&& d_j \end{matrix}\right ), \end{align*}
  where $P_0=(|\mathfrak{t}|,0)$ , $P_1=(\lfloor{|\mathfrak{t}|-1}/{2}\rfloor,1)$ , $\delta =\delta _{V}=1$ and $k_j=k_{j+1}=0$ .

• If $\mathfrak{t}$ is even, then for $V=V_{\mathfrak{t}}^{w_h}$ and for all $j=0,\dots, r_{V}$ , the matrices $M_{V,j}$ and $M_{V,j}^{-1}$ are respectively

  \begin{align*} \left (\begin{matrix} -\delta \frac{|\mathfrak{t}|}{2} && -\left (\frac{|\mathfrak{t}|}{2}+1\right ) d_j - k_j\frac{|\mathfrak{t}|}{2} && \left (\frac{|\mathfrak{t}|}{2}+1\right ) d_{j+1}+k_{j+1}\frac{|\mathfrak{t}|}{2}\\[4pt] \delta && d_j+k_j &&-d_{j+1}-k_{j+1}\\[4pt] -\delta \frac{\epsilon _{\mathfrak{t}}-|\mathfrak{t}|\rho _{\mathfrak{t}}}{2} && \frac{n_j}{\delta }-k_j\frac{\epsilon _{\mathfrak{t}}-|\mathfrak{t}|\rho _{\mathfrak{t}}}{2} &&-\frac{n_{j+1}}{\delta }+k_{j+1}\frac{\epsilon _{\mathfrak{t}}-|\mathfrak{t}|\rho _{\mathfrak{t}}}{2} \end{matrix}\right ), \end{align*}
  \begin{align*} T_j\cdot \left (\begin{matrix} 1 && \frac{|\mathfrak{t}|}{2}+1 && 0\\[4pt] d_{j+1}\rho _{\mathfrak{t}} -\gamma _{j+1} && \frac{d_{j+1}\epsilon _{\mathfrak{t}}}{2}-\frac{|\mathfrak{t}|}{2}\gamma _{j+1}&& d_{j+1}\\[4pt] d_j\rho _{\mathfrak{t}} -\gamma _{j} && \frac{d_j\epsilon _{\mathfrak{t}}}{2} -\frac{|\mathfrak{t}|}{2}\gamma _{j}&& d_j \end{matrix}\right ), \end{align*}
  where $P_0=(|\mathfrak{t}|,0)$ , $P_1=(\lfloor{|\mathfrak{t}|-1}/{2}\rfloor,1)$ and $\delta =\delta _{V}$ .

• If $f(w_h)=0$ , then for $V_0=V_0^{w_h}$ and for all $j=0,\dots, r_{V_0}$ , the matrices $M_{V_0,j}$ and $M_{V_0,j}^{-1}$ are respectively

  \begin{align*} \left (\begin{matrix} 1 && d_j && -d_{j+1}\\[4pt] -2 && -d_j &&d_{j+1}\\[4pt] \epsilon _{{\mathfrak{s}}_h}-\rho _{{\mathfrak{s}}_h} && n_j &&-n_{j+1} \end{matrix}\right ),\qquad T_j\cdot \left (\begin{matrix} -1 && -1 && 0\\[4pt] d_{j+1}\rho _{{\mathfrak{s}}_h} +2\cdot \gamma _{j+1} && \frac{d_{j+1}\epsilon _{{\mathfrak{s}}_h}}{2}+\gamma _{j+1}&& d_{j+1}\\[4pt] d_j\rho _{{\mathfrak{s}}_h} +2\cdot \gamma _{j} && \frac{d_j\epsilon _{{\mathfrak{s}}_h}}{2}+\gamma _{j}&& d_j \end{matrix}\right ), \end{align*}
  where $P_0=(0,2)$ , $P_1=(1,1)$ , $\delta =\delta _{V_0}=1$ and $k_j=k_{j+1}=0$ .

• If $f(w_h)\neq 0$ , then for $V_0=V_0^{w_h}$ and for all $j=0,\dots, r_{V_0}$ , the matrices $M_{V_0,j}$ and $M_{V_0,j}^{-1}$ are respectively

  \begin{align*} \left (\begin{matrix} 0 && d_j && - d_{j+1}\\[4pt] -\delta && -d_j-k_j &&d_{j+1}+k_{j+1}\\[4pt] \delta \frac{\epsilon _{{\mathfrak{s}}_h}}{2} && \frac{n_j}{\delta }+k_j\frac{\epsilon _{{\mathfrak{s}}_h}}{2} &&-\frac{n_{j+1}}{\delta }-k_{j+1}\frac{\epsilon _{{\mathfrak{s}}_h}}{2} \end{matrix}\right ),\qquad T_j\cdot \left (\begin{matrix} -1 && -1 && 0\\[4pt] d_{j+1}\rho _{{\mathfrak{s}}_h} +\gamma _{j+1} && \frac{d_{j+1}\epsilon _{{\mathfrak{s}}_h}}{2}&& d_{j+1}\\[4pt] d_j\rho _{{\mathfrak{s}}_h} +\gamma _{j} && \frac{d_j\epsilon _{{\mathfrak{s}}_h}}{2}&& d_j \end{matrix}\right ), \end{align*}
  where $P_0=(0,2)$ , $P_1=(1,1)$ and $\delta =\delta _{V_0}$ .

Proof. We follow the notation of [1, §4]. Choose $P_0,P_1\in \mathbb{Z}^2$ as in the proof of Lemma 4.3.

First consider the edge $L_{\mathfrak{t}}^{w_h}$ of $F_{\mathfrak{t}}^{w_h}$ . From Lemma 4.3 we have

\begin{align*} \nu =(1,0,-\rho _t)\quad \mbox{ and }\quad (w_x,w_y)=\left (-\left \lfloor |\mathfrak{t}|/2\right \rfloor -1,1\right ). \end{align*}

Then $M_{L_{\mathfrak{t}}^{w_h},i}$ and $M_{L_{\mathfrak{t}}^{w_h},i}^{-1}$ follow from [1, §4.3] as $k_i\equiv n_i(\delta \rho _{\mathfrak{t}})^{-1}\mod \delta$ and

\begin{align*} \frac{n_0}{\delta d_0}=\frac{1}{\delta }s_1^{L_{\mathfrak{t}}^{w_h}}=v_{F_{\mathfrak{t}}^{w_h}}(P_1)-v_{F_{\mathfrak{t}}^{w_h}}(P_0)=-\frac{\epsilon _{\mathfrak{t}}}{2}+\left (\left \lfloor |\mathfrak{t}|/2\right \rfloor +1\right )\rho _{\mathfrak{t}} \end{align*}

Now assume $\mathfrak{t}$ even and consider the edge $V_{\mathfrak{t}}^{w_h}$ of $F_{\mathfrak{t}}^{w_h}$ . Since $\mathfrak{t}$ is even,

\begin{align*} V_{\mathfrak{t}}^{w_h}(\mathbb{Z})=\left \{(|\mathfrak{t}|,0),\left ( \frac{|\mathfrak{t}|}{2},1\right ),(0,2)\right \},\quad \nu =\left (-\frac{|\mathfrak{t}|}{2},1,-\frac{\epsilon _t}{2}+\frac{|\mathfrak{t}|}{2}\rho _{\mathfrak{t}}\right ) \end{align*}

and $(w_x,w_y)=\left (-\frac{|\mathfrak{t}|}{2}-1,1\right )$ as above. Then $M_{V_{\mathfrak{t}}^{w_h},j}$ and $M_{V_{\mathfrak{t}}^{w_h},j}^{-1}$ follow again from [1, (4.3)] as

\begin{align*} \frac{n_0}{\delta d_0}=\frac{1}{\delta }s_1^{V_{\mathfrak{t}}^{w_h}}=v_{F_{\mathfrak{t}}^{w_h}}(P_1)-v_{F_{\mathfrak{t}}^{w_h}}(P_0)=-\frac{\epsilon _{\mathfrak{t}}}{2}+\left (\frac{|\mathfrak{t}|}{2}+1\right )\rho _{\mathfrak{t}}. \end{align*}

Similar arguments and computations yield the remaining matrices.

Remark 5.3. From the Lemma above one can explicitly construct the charts of the model $\mathcal{C}_\Delta ^{w_h}$ . The description of its special fibre $\mathcal{C}_{\Delta,s}^{w_h}$ which follows from [ 1, Theorem 3.14], matches the one given in Theorem 4.18 in the case $W=\{w_h\}$ .

5.2. Open subschemes

Let $h=1,\dots,m$ and let $\mathfrak{t}\in \Sigma _C^{w_h}$ be a proper cluster. Let $M$ be a matrix associated to $\mathfrak{t}$ . Write

\begin{align*} M=\begin{pmatrix} m_{11} && m_{12} && m_{13}\\[4pt] m_{21} && m_{22} && m_{23}\\[4pt] m_{31} && m_{32} && m_{33} \end{pmatrix}\qquad \mbox{and}\qquad M^{-1}=\begin{pmatrix} \tilde{m}_{11} &&\tilde{m}_{12} && \tilde{m}_{13}\\[4pt] \tilde{m}_{21} && \tilde{m}_{22} && \tilde{m}_{23}\\[4pt] \tilde{m}_{31} && \tilde{m}_{32} & &\tilde{m}_{33} \end{pmatrix} \end{align*}

Recall that $X_M=\textrm{Spec}\, R$ , where

\begin{align*} R=\frac{O_K[X^{\pm 1},Y,Z]}{\left ( \pi -X^{\tilde{m}_{13}}Y^{\tilde{m}_{23}}Z^{\tilde{m}_{33}}\right )}\hookrightarrow \frac{O_K[X^{\pm 1},Y^{\pm 1},Z^{\pm 1}]}{\left ( \pi -X^{\tilde{m}_{13}}Y^{\tilde{m}_{23}}Z^{\tilde{m}_{33}}\right )}\stackrel{M}{\simeq } K\left [x^{\pm 1},y^{\pm 1}\right ], \end{align*}

via the change of variable

\begin{align*} \left (\begin{matrix}X\\ Y\\ Z\\\end{matrix}\right )=\left (\begin{matrix}x^{m_{11}}y^{m_{21}}\pi ^{m_{31}}\\ x^{m_{12}}y^{m_{22}}\pi ^{m_{32}}\\ x^{m_{13}}y^{m_{23}}\pi ^{m_{33}}\end{matrix}\right )=\left (\begin{matrix}x\\ y\\ z\end{matrix}\right )\bullet M, \quad \left (\begin{matrix}x\\ y\\ \pi \end{matrix}\right )=\left (\begin{matrix}X^{\tilde{m}_{11}}Y^{\tilde{m}_{21}}Z^{\tilde{m}_{31}}\\ X^{\tilde{m}_{12}}Y^{\tilde{m}_{22}}Z^{\tilde{m}_{32}}\\ X^{\tilde{m}_{13}}Y^{\tilde{m}_{23}}Z^{\tilde{m}_{33}}\end{matrix}\right )=\left (\begin{matrix}X\\ Y\\ Z\end{matrix}\right )\bullet M^{-1}. \end{align*}

Let $l\neq h$ . Set

\begin{align*} T_M^{hl}(X,Y,Z)\;:\!=\;\begin{cases}1+u_{hl}X^{\rho _{hl}\tilde{m}_{13}-\tilde{m}_{11}}Y^{\rho _{hl}\tilde{m}_{23}-\tilde{m}_{21}}Z^{\rho _{hl}\tilde{m}_{33}-\tilde{m}_{31}}&\mbox{if }\mathfrak{t}\supseteq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l,\\ u_{hl}^{-1}X^{\tilde{m}_{11}-\rho _{hl}\tilde{m}_{13}}Y^{\tilde{m}_{21}-\rho _{hl}\tilde{m}_{23}}Z^{\tilde{m}_{31}-\rho _{hl}\tilde{m}_{33}}+1&\mbox{if }\mathfrak{t}\not \supseteq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l, \end{cases} \end{align*}

element of $R[Y^{-1},Z^{-1}]$ . Note that

\begin{align*} &\mbox{if}\quad \mathfrak{t}\supseteq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l\quad \mbox{then}\quad T_M^{hl}(X,Y,Z)\stackrel{M}{\longmapsto }\frac{x+w_{hl}}{x},\\&\mbox{if}\quad \mathfrak{t}\not \supseteq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l\quad \mbox{then}\quad T_M^{hl}(X,Y,Z)\stackrel{M}{\longmapsto }\frac{x+w_{hl}}{w_{hl}}. \end{align*}

The following two lemmas prove that $T_M^{hl}(X,Y,Z)\in R$ . Therefore, up to units, $T_M^{hl}(X,Y,Z)$ can be seen as the polynomial in $O_K[X^{\pm 1},Y,Z]$ satisfying

\begin{align*} x-w_{hl}\stackrel{M}{=}X^{n_X} Y^{n_Y} Z^{n_Z}T_M^{hl}(X,Y,Z), \end{align*}

with $n_X,n_Y,n_Z\in \mathbb{Z}$ , such that $\textrm{ord}_Y(T_M^{hl})=\textrm{ord}_Z(T_M^{hl})=0$ .

Lemma 5.4. Let $h,l=1,\dots,m$ , with $h\neq l$ , let $t\in \Sigma _C^{w_h}$ be such that $\mathfrak{t}\supseteq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ and let $M$ be a matrix associated to $\mathfrak{t}$ . Then

\begin{align*} \rho _{hl} \tilde{m}_{23}-\tilde{m}_{21}\geq \rho _{\mathfrak{t}} \tilde{m}_{23}-\tilde{m}_{21}\geq 0 \quad \mbox{and}\quad \rho _{hl} \tilde{m}_{33}- \tilde{m}_{31}\geq \rho _{\mathfrak{t}} \tilde{m}_{33}-\tilde{m}_{31}\geq 0. \end{align*}

Furthermore if $M=M_{L_{\mathfrak{t}}^{w_h},i}$ then

• $\rho _{hl} \tilde{m}_{23}-\tilde{m}_{21}=0$ if and only if $i=r_{L_{\mathfrak{t}}^{w_h}}$ or $\mathfrak{t}={\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ ,

• $\rho _{hl} \tilde{m}_{33}-\tilde{m}_{31}=0$ if and only if $\mathfrak{t}={\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ ;

if $M=M_{V_{\mathfrak{t}}^{w_h},j}$ then

• $\rho _{hl} \tilde{m}_{23}-\tilde{m}_{21}\gt 0$ ,

• $\rho _{hl} \tilde{m}_{33}-\tilde{m}_{31}=0$ if and only if $\mathfrak{t}={\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ and $j=0$ .

Proof. This result follows from Lemma 5.2, which gives a complete description of $M$ and $M^{-1}$ . We show it when $\mathfrak{t}$ is even and $M=M_{V_{\mathfrak{t}}^{w_h},j}$ , and leave the other cases for the reader. First of all recall that $\rho _{hl}=\rho _{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}$ by Lemma 3.18. Then

\begin{align*} \rho _{hl} \tilde{m}_{23}-\tilde{m}_{21}= \delta \left ( d_{j+1}\left (\rho _{hl}-\rho _t\right )+\gamma _{j+1}\right )\gt \delta d_{j+1}\left (\rho _{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}-\rho _{\mathfrak{t}}\right )\geq 0, \end{align*}

where $\gamma _{j}=\tfrac{n_0d_j-n_jd_0}{\delta d_0}$ and $\delta =\delta _M$ . Similarly,

\begin{align*} \rho _{hl} \tilde{m}_{33}-\tilde{m}_{31}= \delta \left ( d_{j}\left (\rho _{hl}-\rho _t\right )+\gamma _j\right )\geq \delta d_{j}\left (\rho _{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}-\rho _{\mathfrak{t}}\right )\geq 0. \end{align*}

In particular $\rho _{hl} \tilde{m}_{33}-\tilde{m}_{31}=0$ if and only if $\mathfrak{t}={\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ and $j=0$ .

Lemma 5.5. Let $\mathfrak{t}\in \Sigma _C^{w_h}$ be a proper cluster such that $\mathfrak{t}\not \supseteq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ , and let $M$ be a matrix associated to $\mathfrak{t}$ . Then

\begin{align*} \tilde{m}_{21}-\rho _{hl} \tilde{m}_{23}\geq 0\quad \mbox{and}\quad \tilde{m}_{31}-\rho _{hl} \tilde{m}_{33}\gt 0. \end{align*}

Furthermore, $\tilde{m}_{21}-\rho _{hl} \tilde{m}_{23}=0$ if and only if

• $M=M_{L_{\mathfrak{t}}^{w_h},i}$ and $i=r_{L_{\mathfrak{t}}^{w_h}}$ , or

• $\mathfrak{t}\lt{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ , $M=M_{V_{\mathfrak{t}}^{w_h},j}$ , and $j=r_{V_{\mathfrak{t}}^{w_h}}$ .

Proof. This result follows again from Lemma 5.2. As in the previous lemma, we show it when $\mathfrak{t}$ is even and $M=M_{V_{\mathfrak{t}}^{w_h},j}$ , and leave the other cases for the reader.

Let $r=r_{V_{\mathfrak{t}}^{w_h}}$ . Note that $t\neq \mathfrak{R}$ . Set $\delta =\delta _M$ and $\gamma _j=\tfrac{n_0d_j-n_jd_0}{\delta d_0}$ . Then

\begin{align*} \tilde{m}_{31}-\rho _{hl} \tilde{m}_{33}= \delta \left ( d_{j}\left (\rho _t-\rho _{hl}\right )-\gamma _j\right )\gt \delta d_{j}\left (\rho _{P(\mathfrak{t})}-\rho _{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}\right )\geq 0. \end{align*}

since $d_j\gt 0$ and $\gamma _j/d_j\lt \gamma _{r+1}/d_{r+1}=\rho _{\mathfrak{t}}-\rho _{P(\mathfrak{t})}.$ Similarly,

\begin{align*} \tilde{m}_{21}-\rho _{hl} \tilde{m}_{23}= \delta \left ( d_{j+1}\left (\rho _t-\rho _{hl}\right )-\gamma _{j+1}\right )\geq \delta d_{j+1}\left (\rho _{P(\mathfrak{t})}-\rho _{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}\right )\geq 0, \end{align*}

In particular $\tilde{m}_{21}-\rho _{hl} \tilde{m}_{23}=0$ if and only if $\mathfrak{t}\lt{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ and $j=r$ .

Let

\begin{align*} T_M^h(X,Y,Z)\;:\!=\;\prod _{l\neq h}T_M^{hl}(X,Y,Z), \end{align*}

and define

\begin{align*} V_M^h\;:\!=\;\textrm{Spec}\, R[T_M^h(X,Y,Z)^{-1}]\subset X_M,\quad \mbox{ and }\quad{X}_\Delta ^h\;:\!=\;\bigcup _{\mathfrak{t},M} V_M^h\subseteq X_\Delta ^h, \end{align*}

where $\mathfrak{t}$ runs through all proper clusters in $\Sigma _C^{w_h}$ and $M$ runs through all matrices associated to $\mathfrak{t}$ . We can then define the subscheme

\begin{align*}\mathring{\mathcal{C}}_\Delta ^{w_h}\;:\!=\; \mathcal{C}_{\Delta }^{w_h}\cap{X}_\Delta ^h\subset X_\Delta ^h, \end{align*}

where $\mathcal{C}_{\Delta }^{w_h}/O_K$ is the model of the hyperelliptic curve $C^{w_h}:y^2=f(x+w_h)$ described in [1, Theorem 3.14] (see [1, §4] for the construction). Explicitly, let $g_h(x,y)\;:\!=\;y^2-f(x+w_h)$ and define $\mathcal{F}_M^h\in O_K[X^{\pm 1},Y,Z]$ such that $\textrm{ord}_Y(\mathcal{F}_M^h)=\textrm{ord}_Z(\mathcal{F}_M^h)=0$ , with all non-zero coefficients in $O_K^\times$ , satisfying

\begin{align*} y^2-f(x+w_h)\stackrel{M}{=}Y^{n_{Y,h}}Z^{n_{Z,h}}\mathcal{F}_M^h(X,Y,Z), \end{align*}

for unique $n_{Y,h},n_{Z,h}\in \mathbb{Z}$ . Consider the subscheme

\begin{align*} U_M^h\;:\!=\;\textrm{Spec}\,\frac{R\left [T_M^h(X,Y,Z)^{-1}\right ]}{\left (\mathcal{F}_M^h(X,Y,Z)\right )}\subset V_M^h. \end{align*}

Then

\begin{align*}\mathring{\mathcal{C}}_\Delta ^{w_h}=\bigcup _{\mathfrak{t},M} U_M^h\subset{X}_\Delta ^h, \end{align*}

where $\mathfrak{t}$ runs through all proper clusters in $\Sigma _C^{w_h}$ and $M$ runs through all matrices associated to $\mathfrak{t}$ , as before.

5.3. Glueing

Let $h,l=1,\dots,m$ , with $h\neq l$ . Consider the isomorphism

(1) \begin{equation} \phi :K\left [x^{\pm 1},y^{\pm 1},\prod _{o\neq l}(x+w_{lo})^{-1}\right ]\stackrel{\simeq }{\longrightarrow } K\left [x^{\pm 1},y^{\pm 1},\prod _{o\neq h}(x+w_{ho})^{-1}\right ] \end{equation}

sending $x\mapsto x+w_{hl}$ , $y\mapsto y$ . If $\mathfrak{t}\supseteq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ and $M$ is a matrix associated to $\mathfrak{t}$ , then $\phi$ gives a map

\begin{align*} R[Y^{-1},Z^{-1},T_M^l(X,Y,Z)^{-1}]\xrightarrow{M^{-1}\circ \phi \circ M}R[Y^{-1},Z^{-1},T_M^h(X,Y,Z)^{-1}], \end{align*}

which sends

\begin{align*} F(X,Y,Z)\mapsto F(X\cdot T_M^{hl}(X,Y,Z)^{m_{11}}, Y\cdot T_M^{hl}(X,Y,Z)^{m_{12}},Z\cdot T_M^{hl}(X,Y,Z)^{m_{13}}). \end{align*}

Hence it induces the isomorphisms

(2) \begin{equation} R[T_M^l(X,Y,Z)^{-1}]\stackrel{\simeq }{\longrightarrow } R[T_M^h(X,Y,Z)^{-1}], \qquad V_M^h\stackrel{\simeq }{\longrightarrow }V_M^l. \end{equation}

Via these maps we see that $g_h(x,y)=Y^{n_{Y,h}}Z^{n_{Z,h}}\mathcal{F}_M^h(X,Y,Z)$ also equals

\begin{align*} Y^{n_{Y,l}}\cdot Z^{n_{Z,l}}\cdot (T_M^{hl})^{n_{Y,l}m_{12}+n_{Z,l}m_{13}}\mathcal{F}_M^l\left ( X\cdot (T_M^{hl})^{m_{11}},Y\cdot (T_M^{hl})^{m_{12}},Z\cdot (T_M^{hl})^{m_{13}}\right ), \end{align*}

where $T_M^{hl}=T_M^{hl}(X,Y,Z)$ . Since neither $Y$ nor $Z$ divide $T_M^{hl}(X,Y,Z)$ , we have $n_{Y,h}=n_{Y,l}$ , $n_{Z,h}=n_{Z,l}$ and

\begin{align*} \mathcal{F}_M^h(X,Y,Z)=(T_M^{hl})^{n_{Y,l}m_{12}+n_{Z,l}m_{13}}\mathcal{F}_M^l\left ( X\, (T_M^{hl})^{m_{11}},Y\, (T_M^{hl})^{m_{12}},Z\, (T_M^{hl})^{m_{13}}\right ). \end{align*}

Hence (2) induces the isomorphisms

(3) \begin{equation} \frac{R\left [T_M^l(X,Y,Z)^{-1}\right ]}{\left (\mathcal{F}_M^l(X,Y,Z)\right )}\stackrel{\simeq }{\longrightarrow }\frac{R\left [T_M^h(X,Y,Z)^{-1}\right ]}{\left (\mathcal{F}_M^h(X,Y,Z)\right )},\qquad U_M^h\stackrel{\simeq }{\longrightarrow }U_M^l. \end{equation}

Define the subschemes

\begin{align*} V^{hl}\;:\!=\;\bigcup _{\mathfrak{t}_l,M_l}V_{M_l}^{h}\subseteq{X}_{\Delta }^h,\qquad U^{hl}\;:\!=\;V^{hl}\cap \mathcal{C}_{\Delta }^{w_h}\subseteq\mathring{\mathcal{C}}_{\Delta }^{w_h}, \end{align*}

where $\mathfrak{t}_l$ runs through all proper clusters in $\Sigma _C^{w_h}\cap \Sigma _C^{w_l}$ (i.e. $\mathfrak{t}_l\in \Sigma ^W$ , ${\mathfrak{s}}_h\wedge{\mathfrak{s}}_l\subseteq \mathfrak{t}_l$ ) and $M_l$ runs through all matrices associated to $\mathfrak{t}_l$ . From (1), (2) and (3) we have isomorphisms of schemes

(4) \begin{equation} V^{hl}\stackrel{\simeq }{\longrightarrow }V^{lh},\qquad U^{hl}\stackrel{\simeq }{\longrightarrow }U^{lh}. \end{equation}

Now, $U^{hl}\subset V^{hl}$ are open subschemes respectively of $\mathring{\mathcal{C}}_\Delta ^{w_h}\subset{X}_\Delta ^h$ for any $l\neq h$ . Glueing the schemes $\mathring{\mathcal{C}}_\Delta ^{w_h}\subset{X}_\Delta ^h$ , to varying of $h=1,\dots,m$ , respectively along the opens $U^{hl}\subset V^{hl}$ via (4) gives the schemes $\mathcal{C}\subset \mathcal{X}$ . We will show that $\mathcal{C}/O_K$ is a proper flat4 model of $C$ .

5.4. Generic fibre

We start studying the generic fibre $\mathcal{C}_\eta$ of $\mathcal{C}$ . Since it is the glueing of all $\mathring{\mathcal{C}}_{\Delta,\eta }^{w_h}$ through the glueing maps

\begin{align*} U^{hl}_\eta \longrightarrow U^{lh}_\eta \end{align*}

induced by (4), we start focusing on $\mathring{\mathcal{C}}_{\Delta,\eta }^{w_h}$ for $h=1,\dots,m$ . In particular, as $\mathring{\mathcal{C}}_\Delta ^{w_h}$ is an open subscheme of $\mathcal{C}_\Delta ^{w_h}$ , we study $\mathcal{C}_{\Delta,\eta }^{w_h}\smallsetminus\mathring{\mathcal{C}}_{\Delta,\eta }^{w_h}=C^{w_h}\smallsetminus\mathring{\mathcal{C}}_{\Delta,\eta }^{w_h}$ .

Lemma 5.6. For any $h=1,\dots,m$ ,

\begin{align*} C^{w_h}\smallsetminus\mathring{\mathcal{C}}_{\Delta,\eta }^{w_h}=\textrm{Spec}\,\frac{K[x,y]}{\left ( g_h(x,y), \prod _{o\neq h}\left ( x+w_{ho}\right )\right )}. \end{align*}

Proof. For every choice of a proper cluster $\mathfrak{t}\in \Sigma _C^{w_h}$ , and $M$ associated to $\mathfrak{t}$ , let

\begin{align*} P_M\;:\!=\;\left ( \mathcal{C}_{\Delta,\eta }^{w_h}\smallsetminus\mathring{\mathcal{C}}_{\Delta,\eta }^{w_h}\right )\cap X_M=\textrm{Spec}\, \frac{R\otimes _{O_K} K}{\left (\mathcal{F}_M^h(X,Y,Z),T_M^h(X,Y,Z)\right )}. \end{align*}

To study $P_M$ we are going to use Lemma 5.2 and the definition of $T_M^h(X,Y,Z)$ .

Suppose first $\mathfrak{t}\neq \mathfrak{R}$ and $M=M_{V_{\mathfrak{t}}^{w_h},j}$ . Then $\tilde{m}_{23},\tilde{m}_{33}\gt 0$ , so

(5) \begin{equation} P_M=\textrm{Spec}\, \frac{R[Y^{- 1}, Z^{- 1}]}{\left (\mathcal{F}_M^h(X,Y,Z),T_M^h(X,Y,Z)\right )}\stackrel{M}{\simeq } \textrm{Spec}\,\frac{K[x^{\pm 1},y^{\pm 1}]}{\left ( g_h(x,y), \prod _{o}\left ( x+w_{ho}\right )\right )}, \end{equation}

where the product runs over all $o\neq h$ . Now let $\mathfrak{t}=\mathfrak{R}$ and $M=M_{V_{\mathfrak{t}}^{w_h},j}$ . If $j\neq r_{V_{\mathfrak{R}}^{w_h}}$ , then $P_M$ is as in the previous case (since $\tilde{m}_{23},\tilde{m}_{33}\gt 0$ ). If $j= r_{V_{\mathfrak{R}}^{w_h}}$ , then $\tilde{m}_{33}\gt 0$ , $\tilde{m}_{23}=0$ , but $\rho _{hl} \tilde{m}_{23}-\tilde{m}_{21}\gt 0$ by Lemma 5.4. So from the definition of $T_M^{hl}(X,Y,Z)$ we have once more the equality (5). Similarly, if $\mathfrak{t}={\mathfrak{s}}_h$ and $M=M_{V_0^{w_h},j}$ , then $\tilde{m}_{33}\gt 0$ , and $\tilde{m}_{21}-\rho _{hl} \tilde{m}_{23}\gt 0$ by Lemma 5.5. Hence we have (5) again.

It remains to study $P_M$ when $M=M_{L_{\mathfrak{t}}^{w_h},i}$ . If $i\neq r_{L_{\mathfrak{t}}^{w_h}}$ , then $\tilde{m}_{23},\tilde{m}_{33}\gt 0$ and so $P_M$ is as in (5). Let $i=r_{L_{\mathfrak{t}}^{w_h}}$ . Then $\tilde{m}_{33}\gt 0$ but both $\tilde{m}_{23}$ and $\rho _{hl} \tilde{m}_{23}-\tilde{m}_{21}$ equal $0$ . Hence $\tilde{m}_{23}=\tilde{m}_{21}=0$ , which also implies $m_{21}=m_{23}=0$ . Therefore $M$ defines an isomorphism $R[Z^{-1}]\simeq K[x^{\pm 1},y],$ which induces

\begin{align*} P_M=\textrm{Spec}\, \frac{R[Z^{- 1}]}{\left (\mathcal{F}_M^h(X,Y,Z),T_M^h(X,Y,Z)\right )}\stackrel{M}{\simeq } \textrm{Spec}\,\frac{K[x^{\pm 1},y]}{\left ( g_h(x,y), \prod _{o\neq h}\left ( x+w_{ho}\right )\right )}. \end{align*}

This concludes the proof.

Regarding $\mathcal{C}_\Delta ^{w_h}$ as a model of $C$ via the natural isomorphism $C\xrightarrow{\sim }C^{w_h}$ , we get

\begin{align*} C\smallsetminus\mathring{\mathcal{C}}_{\Delta,\eta }^{w_h}=\textrm{Spec}\,\frac{K[x,y]}{\left ( y^2-f(x), \prod _{o\neq h}\left ( x-w_o\right )\right )}. \end{align*}

Thus the generic fibre of $\mathcal{C}$ is isomorphic to $C$ .

5.5. Special fibre

We now study the structure of the special fibre $\mathcal{C}_s$ of $\mathcal{C}$ . As for the generic fibre, we consider

\begin{align*} \mathcal{C}_{\Delta,s}^{w_h}\smallsetminus\mathring{\mathcal{C}}_{\Delta,s}^{w_h}, \end{align*}

for any $h=1,\dots,m$ . For every choice of a proper cluster $\mathfrak{t}\in \Sigma _C^{w_h}$ , and $M$ associated to $\mathfrak{t}$ , let

\begin{align*} S_M\;:\!=\;\left ( \mathcal{C}_{\Delta,s}^{w_h}\smallsetminus\mathring{\mathcal{C}}_{\Delta,s}^{w_h}\right )\cap X_M=\textrm{Spec}\, \frac{O_K[X^{\pm 1},Y,Z]}{\left (\mathcal{F}_M^h(X,Y,Z),T_M^h(X,Y,Z),Y^{\tilde{m}_{23}}Z^{\tilde{m}_{33}},\pi \right )}. \end{align*}

Lemma 5.7. Let $M=M_{L,i}$ for $L=L_{\mathfrak{t}}^{w_h}$ . Let $l\neq h$ . If $\mathfrak{t}={\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ , then $T_M^{hl}(X,Y,Z)=X^{-1}(X+u_{hl})$ , otherwise

1. (i) $T_M^{hl}(X,Y,0)=1$ for $i=0,\dots,r_L$ ;

2. (ii) $T_M^{hl}(X,0,Z)=1$ for $i=0,\dots,r_L-1$ .

Proof. Fix $l\neq h$ . If $\mathfrak{t}\not \supseteq{\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ , then by Lemma 5.5, we have $\tilde{m}_{21}-\rho _{hl} \tilde{m}_{23}\geq 0$ and $\tilde{m}_{31}-\rho _{hl} \tilde{m}_{33}\gt 0$ . Moreover, if $\tilde{m}_{21}-\rho _{hl} \tilde{m}_{23}=0$ , then $i=r_L$ . Therefore the equalities in (i) and (ii) follow directly from the definition of $T_M^{hl}$ .

On the other hand, if $\mathfrak{t}\supsetneq{\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ , then by Lemma 5.4, we have $\rho _{hl} \tilde{m}_{23}-\tilde{m}_{21}\geq 0$ and $\rho _{hl} \tilde{m}_{33}-\tilde{m}_{31}\gt 0$ . Moreover, if $\rho _{hl} \tilde{m}_{23}-\tilde{m}_{21}=0$ , then $i=r_L$ . Therefore we have (i) and (ii) again.

Finally, assume $\mathfrak{t}={\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ . Since $\rho _{\mathfrak{t}}=\rho _{hl}\in \mathbb{Z}$ , then $\rho _{hl} \tilde{m}_{13} - \tilde{m}_{11}=-1$ . Hence

\begin{align*} T_M^{hl}(X,Y,Z)=1+ u_{hl} X^{-1}=X^{-1}\left ( X+u_{hl}\right ), \end{align*}

by Lemma 5.4.

Lemma 5.8. Suppose $M=M_{L_{\mathfrak{t}}^{w_h},i}$ . Then

\begin{align*} S_M=\textrm{Spec}\,{\dfrac{O_K[X^{\pm 1},Y,Z]}{(\mathcal{F}_M^h(X,Y,Z),\prod _l\left ( X+ u_{hl}\right ),Y^{\tilde{m}_{23}}Z^{\tilde{m}_{33}},\pi )}}\subset \mathcal{C}_\Delta ^{w_h}, \end{align*}

where the product runs over all $l\neq h$ such that $\mathfrak{t}={\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ .

Proof. Lemma 5.2 shows that $\tilde{m}_{33}$ is always different from $0$ , while $\tilde{m}_{23}=0$ if and only if $i=r_{L_{\mathfrak{t}}^{w_h}}$ . Thus the result follows from Lemma 5.7.

Lemma 5.9. Let $f_h(x)=f(x+w_h)$ and $l\neq h$ . Let $L_{hl}=L_{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}^{w_h}$ and let $\mathfrak{t}_l\in \Sigma _C^{w_l}$ , $\mathfrak{t}_l\lt{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ . Then $\overline{u_{lh}}$ is a multiple root of $\overline{f_h|_{L_{hl}}}$ of order $|\mathfrak{t}_l|$ .

Conversely, if $\Sigma =\{{\mathfrak{s}}_1,\dots,{\mathfrak{s}}_m\}=\Sigma _C^{\textrm{min}}$ , $C$ has an almost rational cluster picture and $\bar \alpha \in \bar{k}$ is a multiple root of $\overline{f_h|_L}$ for some edge $L$ of $\texttt{NP}(f_h)$ , then $\bar \alpha ={\overline{u_{lh}}}$ and $L=L_{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}^{w_h}$ for some $l\neq h$ .

Proof. For any proper cluster $\mathfrak{s}\in \Sigma _f$ , let $\lambda _{\mathfrak{s}}=\min _{r\in \mathfrak{s}}v(r-w_h)$ . Let $\mathfrak{s}\in \Sigma _C^{w_l}$ , with ${\mathfrak{s}}_l\subseteq \mathfrak{s}\subsetneq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ . Then $w_h$ is not rational centre of $\mathfrak{s}$ , and for every root $r\in \mathfrak{s}$ , one has

\begin{align*} v(r-w_h)=v(r-w_{l}+w_{l}-w_h)=\min \{v(r-w_l), \rho _{hl}\}=\rho _{hl}, \end{align*}

as $v(r-w_l)\geq \rho _{\mathfrak{s}}\gt \rho _{hl}$ . Therefore $\lambda _{\mathfrak{s}}=\rho _{hl}\in \mathbb{Z}$ . In particular, $|\lambda _{\mathfrak{s}}|_p\leq 1$ . Furthermore,

\begin{align*} d_{\mathfrak{s}}\geq \rho _{\mathfrak{s}}\gt \lambda _{\mathfrak{s}}=\rho _{hl}\qquad \text{and}\qquad \frac{r-w_h}{\pi ^{\rho _{hl}}}\equiv \frac{w_{lh}}{\pi ^{\rho _{hl}}}\mod \pi, \end{align*}

and so Theorem 3.24(i) implies that ${\overline{u_{lh}}}=\tfrac{w_{lh}}{\pi ^{\rho _{hl}}}\mod \pi$ is a multiple root of $\overline{f_h|_{L_{hl}}}$ , where $L_{hl}=L_{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}^{w_h}$ .

Let $\mathfrak{t}_l\in \Sigma _C^{w_l}$ , $\mathfrak{t}_l\lt{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ . Since ${\mathfrak{s}}_l\subseteq \mathfrak{t}_l\lt{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ we have

\begin{align*} \mathfrak{t}_l=\left \{r\in \mathfrak{R}\mid{\overline{u_{lh}}}=\tfrac{r-w_h}{\pi ^{\rho _{hl}}}\mod \pi \right \}, \end{align*}

as $v(r-w_l)\gt \rho _{hl}$ if and only if ${\overline{u_{lh}}}=\frac{r-w_h}{\pi ^{\rho _{hl}}}\mod \pi$ . Thus the multiplicity of $\overline{u_{lh}}$ is $|\mathfrak{t}_l|$ by Theorem 3.24(ii).

Now let $\bar{\alpha }$ be a multiple root of $\overline{f_h|_L}$ for some edge $L$ of $\texttt{NP}(f_h)$ and let $\mathfrak{s}\in \Sigma _{f}$ associated to $\bar{\alpha }$ by Theorem 3.24(iii). We want to prove that if $C$ has an almost rational cluster picture and $\Sigma =\Sigma _C^{\textrm{min}}$ , then there exists $l\neq h$ so that $\bar \alpha ={\overline{u_{lh}}}$ . Note first $w_h$ is not a rational centre of $\mathfrak{s}$ . Indeed, if $w_h$ is a rational centre of $\mathfrak{s}$ , then

\begin{align*} |\mathfrak{s}|\gt |\lambda _{\mathfrak{s}}|_p=|\rho _{\mathfrak{s}}|_p,\qquad d_{\mathfrak{s}}\gt \lambda _{\mathfrak{s}}=\rho _{\mathfrak{s}}, \end{align*}

which contradicts the fact that $C$ has an almost rational cluster picture. As $\{{\mathfrak{s}}_1,\dots,{\mathfrak{s}}_m\}=\Sigma _C^{\textrm{min}}$ , we must have that $w_{l}$ is a rational centre of $\mathfrak{s}$ , for some $l\neq h$ . Then ${\mathfrak{s}}_l\subseteq \mathfrak{s}\subsetneq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ . Since $\bar \alpha =\tfrac{r-w_h}{\pi ^{\lambda _{\mathfrak{s}}}}\mod \pi$ for any $r\in \mathfrak{s}$ , from above we have $\bar \alpha ={\overline{u_{lh}}}$ . Finally, $L$ is the edge of $\texttt{NP}(f_h)$ of slope $-\lambda _{\mathfrak{s}}=-\rho _{hl}$ . Thus $L=L_{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}^{w_h}$ .

It remains to compute $S_M$ when $M=M_{V,j}$ , where $V=V_{\mathfrak{t}}^{w_h}$ or $V=V_0^{w_h}$ .

Lemma 5.10. Let $M=M_{V,j}$ for $V=V_{\mathfrak{t}}^{w_h}$ , or $V=V_0^{w_h}$ if $\mathfrak{t}={\mathfrak{s}}_h$ . For any $l\neq h$ we have

1. (i) $T_M^{hl}(X,Y,0)=1$ except when $\mathfrak{t}={\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ and $j=0$ ;

2. (ii) $T_M^{hl}(X,0,Z)=1$ except when $\mathfrak{t}\lt{\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ and $j=r_V$ .

Proof. The Lemma immediately follows from Lemmas 5.4 and 5.5.

Lemma 5.11. Let $M=M_{V,j}$ with $V=V_{\mathfrak{t}}^{w_h}$ , or $V=V_0^{w_h}$ if $\mathfrak{t}={\mathfrak{s}}_h$ . Then $S_M=\varnothing$ .

Proof. For any $l\neq h$ , we want to prove that

(6) \begin{equation} S_M^{hl}\;:\!=\;\{T_M^{hl}(X,Y,Z)=Y^{\tilde{m}_{23}}Z^{\tilde{m}_{33}}=0\}=\varnothing. \end{equation}

Lemma 5.2 shows that $\tilde{m}_{33}$ is always different from $0$ and that $\tilde{m}_{23}=0$ if and only if $j=r_V$ , and $V=V_{\mathfrak{R}}^{w_h}$ or $V=V_0^{w_h}$ . Assume that if $\mathfrak{t}={\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ then $j\neq 0$ and that if $\mathfrak{t}\lt{\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ then $j\neq r_{V}$ . Lemma 5.10 implies (6).

If $\mathfrak{t}={\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ and $j=0$ , then $\rho _{hl} \tilde{m}_{33}-\tilde{m}_{31}= 0$ but $\rho _{hl} \tilde{m}_{23}-\tilde{m}_{21}\gt 0$ . So

\begin{align*} S_M^{hl}=\{T_M^{hl}(X,Y,Z)=Z^{\tilde{m}_{33}}=0\}\subset \textrm{Spec}\, R[Y^{-1}]. \end{align*}

Similarly, if $\mathfrak{t}\lt{\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ and $j=r_{V}$ , then $\tilde{m}_{21}-\rho _{hl} \tilde{m}_{23}= 0$ , however $\tilde{m}_{31}-\rho _{hl} \tilde{m}_{33}\gt 0$ . Then

\begin{align*} S_M^{hl}=\{T_M^{hl}(X,Y,Z)=Y^{\tilde{m}_{23}}=0\}\subset \textrm{Spec}\, R[Z^{-1}]. \end{align*}

In both cases, $S_M^{hl}\subseteq X_{F}$ as sets, where $F=F_{{\mathfrak{s}}_l\wedge{\mathfrak{s}}_h}^{w_h}$ ([1, Definition 3.7]). Let $L=L_{{\mathfrak{s}}_l\wedge{\mathfrak{s}}_h}^{w_h}$ , and let $f_h(x)=f(x+w_h)$ and $g_h(x,y)=y^2-f_h(x)$ . By Lemmas 5.8 and 5.9, one has

\begin{align*} S_M^{hl}\subseteq X_{F}\cap S_{M_{L,0}}=\varnothing, \end{align*}

as $\mathcal{F}_{M_{L,0}}^h(X,Y,0)\mod \pi$ equals $Y^b-X^a{\overline{f_h|_L}}(X)$ , for some $a\in \mathbb{Z}$ , $b=1,2$ (see Lemma 5.17 for more details, whose proof is independent of this result). Thus if $V=V_{\mathfrak{t}}^{w_h}$ and $M=M_{V,j}$ , then $S_M=\varnothing$ .

5.6. Components

Now that we have compared the special fibre of $\mathcal{C}$ with those of the models $\mathcal{C}_{\Delta }^{w_h}$ , let us describe closed subschemes that form it. We will first study closed subschemes forming $\mathring{\mathcal{C}}_{\Delta,s}^{w_h}$ and then how they glue in $\mathcal{C}_s$ .

Let $f_h(x)=f(x+w_h)$ and $g_h(x,y)=y^2-f_h(x)$ . According to [1, Theorem 3.14] the special fibre of $\mathcal{C}_\Delta ^{w_h}$ is formed by:

• Chains of ${\mathbb{P}}_k^1$ s coming from $v$ -edges of $\Delta ^{w_h}$ .

• $1$ -dimensional subschemes coming from $v$ -faces of $\Delta ^{w_h}$ .

More precisely, each $v$ -edge $E$ gives a scheme $X_E\times{\mathbb{P}}_E$ , where ${\mathbb{P}}_E$ is a chain of ${\mathbb{P}}_k^1$ s and $X_E\subset{\mathbb{G}}_{m,k}$ is given by ${\overline{g_h|_E}}=0$ . The multiplicities and the length of ${\mathbb{P}}_E$ can be completely described by the slopes of $E$ . On the other hand, each $v$ -face $F$ gives a proper scheme $\bar{X}_{F}$ containing an open subscheme $X_F\subseteq{\mathbb{G}}_{m,k}^2$ given by ${\overline{g_h|_F}}=0$ . How the previous schemes intersect to form $\mathcal{C}_{\Delta,s}^{w_h}$ is described by [1, Theorem 3.14]. The reader is pointed to [1] for more details.

Definition 5.12. Let $\mathfrak{t}\in \Sigma ^W$ be a proper cluster. For any rational centre $w$ of $\mathfrak{t}$ , let $r_{\mathfrak{t},w}=\tfrac{w-r}{\pi ^{\rho _{\mathfrak{t}}}}$ , $u_{\mathfrak{t},w}=c_f\textstyle \prod _{r\in \mathfrak{R}\setminus \mathfrak{t}} r_{\mathfrak{t},w}$ and $u_{{\mathfrak{s}}_h,w_h}^0=c_f\textstyle \prod _{r\in \mathfrak{R}\setminus \{w_h\}} r_{{\mathfrak{s}}_h,w_h}$ . Define ${\overline{f_{\mathfrak{t},w}^W}},{\overline{g_{\mathfrak{t},w}}}\in k[X]$ , and ${\overline{g_{{\mathfrak{s}}_h,w_h}^0}}\in k[X]$ for any $h=1,\dots,m$ , as follows:

1. (i) Let $u=u_{\mathfrak{t},w}$ . Define $\overline{f_{\mathfrak{t},w}^W}$ by

   \begin{align*} \overline{f_{\mathfrak{t},w}^W}(X^{b_{\mathfrak{t}}})= \tfrac{u}{\pi ^{v(u)}}\prod _{r\in \mathfrak{t}\setminus \bigcup _{\mathfrak{s}\lt \mathfrak{t}}\mathfrak{s}} (X+r_{\mathfrak{t},w})\mod \pi, \end{align*}
   where the union runs through all children $\mathfrak{s}$ of $\mathfrak{t}$ in $\Sigma ^W$ . If $\Sigma =\Sigma _C^{\textrm{min}}$ denote $\overline{f_{\mathfrak{t},w}^W}$ by $\overline{f_{\mathfrak{t},w}}$ .

2. (ii) Let $u=u_{\mathfrak{t},w}$ . Define $\overline{g_{\mathfrak{t},w}}(X)\;:\!=\; X^{p_{\mathfrak{t}}/\gamma _{\mathfrak{t}}} - \tfrac{u}{\pi ^{v(u)}}\mod \pi$ .

3. (iii) Let $u=u_{{\mathfrak{s}}_h,w_h}^0$ . Define $\overline{g_{{\mathfrak{s}}_h,w_h}^0}(X)\;:\!=\;X^{p_{{\mathfrak{s}}_h}^0/\gamma _{{\mathfrak{s}}_h}^0} - \tfrac{u}{\pi ^{v(u)}}\mod \pi$ .

Note that the polynomials defined in Definition 5.12 agree with the ones in Definition 4.14 when $w=w_{\mathfrak{t}}$ .

Lemma 5.13. Let $\mathfrak{s},\mathfrak{t}\in \Sigma _C^{\textrm{rat}}$ , with $\mathfrak{s}\subsetneq \mathfrak{t}$ . Let $w',w$ be rational centres of $\mathfrak{s}$ and $\mathfrak{t}$ respectively, and define ${\overline{u_{w'w}}}=\tfrac{w'-w}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ . Then $\overline{u_{w'w}}$ does not depend on the choice of a rational centre $w'$ of $\mathfrak{s}$ .

Proof. Suppose that $w_1,w_2$ are two rational centres of $\mathfrak{s}$ . Then $v(w_1-w_2)\geq \rho _{\mathfrak{s}}\gt \rho _{\mathfrak{t}}$ , and so the Lemma follows.

Remark 5.14. Let $\mathfrak{t}\in \Sigma _C^{w_h}$ . Let $l=1,\dots,m$ , $l\neq h$ . Then $\mathfrak{t}={\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ if and only if it has a child $\mathfrak{s}\in \Sigma _C^{w_l}\setminus \Sigma _C^{w_h}$ . In particular, if this happens, Lemma 5.13 shows that $\overline{u_{lh}}=\frac{w-w_h}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ for any rational centre $w$ of $\mathfrak{s}$ .

Definition 5.15. Let $\mathfrak{t}\in \Sigma _C^{w_h}$ be a proper cluster. Define $\hat{\mathfrak{t}}^W\;:\!=\;\{\mathfrak{s}\in \Sigma ^W\cup \{\varnothing \}\mid \mathfrak{s}\lt \mathfrak{t}\}$ , where $\varnothing \lt \mathfrak{t}$ only if $\mathfrak{t}$ has no child in $\Sigma ^W$ . If $\varnothing \lt \mathfrak{t}$ then we will say that $w_h$ is the rational centre of $\varnothing$ .

Define ${\mathbb{G}}_{\mathfrak{t},w_h}\;:\!=\;{\mathbb{G}}_{m,k}\setminus \bigcup _l\{{\overline{u_{lh}}}\}$ , where the union runs through all $l\neq h$ such that ${\mathfrak{s}}_l\wedge{\mathfrak{s}}_h=\mathfrak{t}$ . Note that Remark 5.14 shows that ${\mathbb{G}}_{\mathfrak{t},w_h}={\mathbb{A}}_k^1\setminus \bigcup _{\mathfrak{s}\in \hat{\mathfrak{t}}^W}\{{\overline{u_{w_{\mathfrak{s}} w_h}}}\}$ , where ${\overline{u_{w_{\mathfrak{s}} w_h}}}=\frac{w_{\mathfrak{s}}-w_h}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ , and $w_{\mathfrak{s}}$ is any rational centre of $\mathfrak{s}$ .

Let $\mathfrak{t}\in \Sigma _C^{w_h}$ be a proper cluster. Let $V=V_{\mathfrak{t}}^{w_h}$ and $M=M_{V,j}$ . In Section 5.5 we showed the special fibre of $U^h_M$ equals $X_M\cap \mathcal{C}_{\Delta,s}^{w_h}$ . Therefore the components of $\mathring{\mathcal{C}}_{\Delta,s}^{w_h}$ coming from $V$ are the same of those of $\mathcal{C}_{\Delta,s}^{w_h}$ given by the same $v$ -edge. Therefore $V$ gives a closed subscheme $X_V\times{\mathbb{P}}_V$ of $\mathring{\mathcal{C}}_{\Delta,s}^{w_h}$ , where ${\mathbb{P}}_V$ is a chain of ${\mathbb{P}}_k^1$ s and $X_V\;:\;\{\overline{g_h|_{V}}=0\}$ over ${\mathbb{G}}_{m,k}$ . Lemma 4.3 implies that $\overline{g_h|_{V}}=\overline{g_{\mathfrak{t},w_h}}$ .

Let $V_0=V_0^{w_h}$ and $M=M_{V_0,j}$ . Similarly to above, $X_M\cap\mathring{\mathcal{C}}_{\Delta,s}^{w_h}=X_M\cap \mathcal{C}_{\Delta,s}^{w_h}$ and so $V_0$ gives rise to a closed subscheme $X_{V_0}\times{\mathbb{P}}_{V_0}$ of $\mathring{\mathcal{C}}_{\Delta,s}^{w_h}$ , where ${\mathbb{P}}_{V_0}$ is a chain of ${\mathbb{P}}_k^1$ s and $X_{V_0}\;:\;\{\overline{g_h|_{V_0}}=0\}$ over ${\mathbb{G}}_{m,k}$ . Note that $\overline{g_h|_{V_0}}=\overline{g_{{\mathfrak{s}}_h,w_h}^0}$ .

Let $\mathfrak{t}\in \Sigma _C^{w_h}$ be a proper cluster. Let $L=L_{\mathfrak{t}}^{w_h}$ and $M=M_{L,i}$ . By Lemma 5.8, the $v$ -edge $L$ gives a subscheme $X_{L}^W\times{\mathbb{P}}_{L}$ of $\mathring{\mathcal{C}}_{\Delta,s}^{w_h}$ , where ${\mathbb{P}}_{L}$ is a chain of ${\mathbb{P}}_k^1$ s of length $r_L$ and $X_L^W\;:\;\{\overline{g_h|_{L}}=0\}$ in ${\mathbb{G}}_{\mathfrak{t},w_h}$ . Note that $r_L=0$ or $1$ by Lemma 4.3 and $r_L=1$ if and only if $D_{\mathfrak{t}}=1$ . Let $\mathfrak{t}_h\in \Sigma _C^{w_h}$ be the unique child of $\mathfrak{t}$ with rational centre $w_h$ or set $\mathfrak{t}_h=\varnothing$ if $\mathfrak{t}$ has no such child. We will show that

(7) \begin{equation} \overline{g_h|_{L}}(X)=-\prod _{\mathfrak{s}\in \hat{\mathfrak{t}}^W,\,\mathfrak{s}\neq \mathfrak{t}_ h}(X+{\overline{u_{w_{\mathfrak{s}} w_h}}})^{|\mathfrak{s}|}\cdot \overline{f_{\mathfrak{t},w_h}^W}(X). \end{equation}

where ${\overline{u_{w_{\mathfrak{s}} w_h}}}=\frac{w_{\mathfrak{s}}-w_h}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ , and $w_{\mathfrak{s}}$ is any rational centre of $\mathfrak{s}$ .

Suppose $\mathfrak{t}\neq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ for any $l\neq h$ . Equivalently, all children of $\mathfrak{t}$ in $\Sigma ^W$ (at most one) belong to $\Sigma _C^{w_h}$ . Then Lemma 4.3 shows that $\overline{g_h|_{L}}=-\overline{f_{\mathfrak{t},w_h}^W}$ . Suppose now that $\mathfrak{t}={\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ for some $l\neq h$ . In this case $b_{\mathfrak{t}}=1$ . We have

\begin{align*} \frac{{\overline{g_h|_L}}(X)}{\prod _{\mathfrak{s}\in \hat{\mathfrak{t}}^W,\mathfrak{s}\neq \mathfrak{t}_ h}(X+{\overline{u_{w_{\mathfrak{s}} w_h}}})^{|\mathfrak{s}|}}=\bigg (\!\frac{-\tfrac{u}{\pi ^{v(u)}}\prod _{r\in \mathfrak{t}\setminus \mathfrak{t}_h}(X+r_{\mathfrak{t},w_h})}{\prod _{\mathfrak{s}\in \hat{\mathfrak{t}}^W,\mathfrak{s}\neq \mathfrak{t}_h}\prod _{r\in \mathfrak{s}}(X+r_{\mathfrak{t},w_h})}\!\!\mod \pi \!\!\bigg )=-\overline{f_{\mathfrak{t},w_h}^W}(X), \end{align*}

where $r_{\mathfrak{t},w_h}$ and $u=u_{\mathfrak{t},w_h}$ are as in Definition 5.12. Indeed, $\overline{u_{w_{\mathfrak{s}} w_h}}= r_{\mathfrak{t},w_h}\mod \pi$ for every $r\in \mathfrak{s}$ as $v(w_{\mathfrak{s}}-r)\geq \rho _{\mathfrak{s}}\gt \rho _{\mathfrak{t}}$ , and since $b_{\mathfrak{t}}=1$ , Lemma 4.3 implies that

\begin{align*}{\overline{g_h|_L}}(x)=-\tfrac{u}{\pi ^{v(u)}}\textstyle \prod _{r\in \mathfrak{t}\setminus \mathfrak{t}_h}(x+r_{\mathfrak{t},w_h})\mod \pi. \end{align*}

In particular, Remark 5.13 and Lemma 5.9 shows that $(X+{\overline{u_{hl}}})\nmid{\overline{f_{\mathfrak{t},w_h}^W}}(X)$ , for any $l\neq h$ such that ${\mathfrak{s}}_l\wedge{\mathfrak{s}}_h=\mathfrak{t}$ . Moreover, $X\nmid{\overline{f_{\mathfrak{t},w_h}^W}}(X)$ by definition. Therefore the scheme $X_L^W$ is equal to the closed subscheme $X_{\mathfrak{t},w_h}^W\subset{\mathbb{A}}_{k}^1$ given by ${\overline{f_{\mathfrak{t},w_h}^W}}=0$ .

Let $\mathfrak{t}\in \Sigma ^W$ be a proper cluster. For any $h=1,\dots,m$ such that ${\mathfrak{s}}_h\subseteq \mathfrak{t}$ , let $\bar{X}_{F_{\mathfrak{t}}^{w_h}}$ be the $1$ -dimensional closed subscheme of $\mathcal{C}_{\Delta,s}^{w_h}$ given by $F_{\mathfrak{t}}^{w_h}$ . Define

\begin{align*}\mathring{X}_{F_{\mathfrak{t}}^{w_h}}\;:\!=\;\bar{X}_{F_{\mathfrak{t}}^{w_h}}\cap\mathring{\mathcal{C}}_{\Delta }^{w_h}. \end{align*}

Denote by $\Gamma _{\mathfrak{t}}$ the $1$ -dimensional closed subscheme of $\mathcal{C}_s$ , result of the glueing of the subschemes $\mathring{X}_{F_{\mathfrak{t}}^{w_h}}$ of $\mathring{\mathcal{C}}_{\Delta,s}^{w_h}$ to varying of $h$ such that $\mathfrak{t}\in \Sigma _C^{w_h}$ .

Lemma 5.16. Let $\mathfrak{t}\in \Sigma _C^{w_h}$ be a proper cluster. The multiplicity of $\Gamma _{\mathfrak{t}}$ in $\mathcal{C}_s$ is $m_{\mathfrak{t}}$ .

Proof. Let $L=L_{\mathfrak{t}}^{w_h}$ , $M=M_{L,0}$ , and let $F=F_{\mathfrak{t}}^{w_h}$ . The multiplicity of $\bar{X}_{F_{\mathfrak{t}}^{w_h}}$ , and so of $ X_{F_{\mathfrak{t}}^{w_h}}$ and $\Gamma _{\mathfrak{t}}$ , is $\delta _F$ . Hence we only need to show that $m_{\mathfrak{t}}=\delta _F$ . Let $d_0\in \mathbb{Z}$ as in Lemma 5.2. Then $\delta _F=\delta _L d_0$ . The result follows as $\delta _L=b_{\mathfrak{t}}$ and $d_0$ , denominator of $s_1^L$ , equals $3-D_{\mathfrak{t}}$ by Lemma 4.3.

Lemma 5.17. Let $L=L_{\mathfrak{t}}^{w_h}$ , $F=F_{\mathfrak{t}}^{w_h}$ and $M=M_{L,0}$ . Let $c\in \{0,\dots,b_{\mathfrak{t}}-1\}$ such that $1/b_{\mathfrak{t}}-\rho _{\mathfrak{t}}\cdot c\in \mathbb{Z}$ . Then $\mathcal{F}_M^h(X,Y,0)\mod \pi$ equals the polynomial

\begin{align*}{\overline{g_h|_F}}(X,Y)=Y^{D_{\mathfrak{t}}}-\prod _{\mathfrak{s}\in \hat{\mathfrak{t}}^W}(X-{\overline{u_{w_{\mathfrak{s}} w_h}}})^{\frac{|\mathfrak{s}|}{b_{\mathfrak{t}}}-c\epsilon _{\mathfrak{t}}}{\overline{f_{\mathfrak{t},w_h}^W}}(X), \end{align*}

where ${\overline{u_{w_{\mathfrak{s}} w_h}}}=\frac{w_{\mathfrak{s}}-w_h}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ , and $w_{\mathfrak{s}}$ is any rational centre of $\mathfrak{s}$ .

In particular, $\Gamma _{\mathfrak{t}}^h\subset{\mathbb{G}}_{\mathfrak{t},w_h}\times{\mathbb{A}}_k^1$ given by ${\overline{g|_F}}=0$ is the open subscheme $U_M^h\cap \{Z=0\}$ of $\mathring{X}_F$ , and the points in $S_M$ belong to all irreducible components of $\bar{X}_{F}$ .

Proof. From [1, §3.5] and the equation of $C^{w_h}$ , the polynomial $\mathcal{F}_M^h(X,Y,0)$ reduces modulo $\pi$ to $X^{a_1}Y^b+X^{a_2}{\overline{g_h|_{L}}}(X)$ , for some $b=1,2$ and $a\in \mathbb{Z}$ . Lemma 4.9 shows that $b=D_{\mathfrak{t}}$ . By Lemma 4.3, $a_1=2\tilde{m}_{12}$ , $a_2=|\mathfrak{t}_h|\tilde{m}_{11}+(\epsilon _{\mathfrak{t}}-|\mathfrak{t}_h|\rho _{\mathfrak{t}})\tilde{m}_{13}$ , where $\mathfrak{t}_h\in \Sigma _C^{w_h}\cup \{\varnothing \}$ , $\mathfrak{t}_h\lt \mathfrak{t}$ . Then $a_1=0$ and $a_2=\tfrac{|\mathfrak{t}_h|}{b_{\mathfrak{t}}}-c\epsilon _{\mathfrak{t}}$ by Lemma 5.2.

If $\mathfrak{t}$ has one or no child, or $D_{\mathfrak{t}}=1$ , then ${\overline{g_h|_L}}=-{\overline{f_{\mathfrak{t},w_h}^W}}$ by (7). On the other hand, if $D_{\mathfrak{t}}=2$ and $\mathfrak{t}$ has two or more children in $\Sigma _C^{\textrm{rat}}$ , then $b_{\mathfrak{t}}=1$ , and so $c=0$ . Therefore the equality (7) concludes the proof of the first part of the statement also in this case. Finally, the last part of the Lemma follows from Lemma 5.8.

Let $c$ as in the previous Lemma and define $\tilde{\mathfrak{t}}^W\;:\!=\;\{\mathfrak{s}\in \hat{\mathfrak{t}}^W\mid \frac{|\mathfrak{s}|}{b_{\mathfrak{t}}}-c\epsilon _{\mathfrak{t}}\notin 2\mathbb{Z}\}$ .

Proposition 5.18. Let $L=L_{\mathfrak{t}}^{w_h}$ and $M=M_{L,0}$ . The dense open subscheme $\Gamma _{\mathfrak{t}}\cap U_M^h$ of $\Gamma _{\mathfrak{t}}$ is isomorphic to the closed subscheme of ${\mathbb{G}}_{\mathfrak{t},w_h}\times{\mathbb{A}}^1_k$ given by

\begin{align*} Y^{D_{\mathfrak{t}}}=\prod _{\mathfrak{s}\in \tilde{\mathfrak{t}}^W}(X-{\overline{u_{w_{\mathfrak{s}} w_h}}})\cdot{\overline{f_{\mathfrak{t},w_h}^W}}(X), \end{align*}

where ${\overline{u_{w_{\mathfrak{s}} w_h}}}=\frac{w_{\mathfrak{s}}-w_h}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ , and $w_{\mathfrak{s}}$ is any rational centre of $\mathfrak{s}$ .

Proof. The proposition follows from Lemma 5.17 and the definition of ${\mathbb{G}}_{\mathfrak{t},w_h}$ .

We conclude this subsection describing how the glueing morphism (4) restricts to the special fibre. Suppose $\mathfrak{t}\supseteq{\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ for $l\neq h$ and let $M$ be a matrix associated to $\mathfrak{t}$ . Consider the glueing map $U_M^h\rightarrow U_M^l$ explicitly defined in Section 5.3.

Suppose first $M=M_{V,j}$ with $V=V_{\mathfrak{t}}^{w_l}$ . By Lemma 5.10 the glueing morphism restricts to the identity on $X_V\times{\mathbb{P}}_V$ .

Suppose $M=M_{L,i}$ with $L=L_{\mathfrak{t}}^{w_l}$ . Note that $\tilde{m}_{12}=0$ from Lemma 5.2. Recall the open subscheme $\Gamma _{\mathfrak{t}}^h$ of $ X_{F_{\mathfrak{t}}^{w_h}}$ defined in Lemma 5.17. Then, Lemma 5.7 implies that the glueing map restricts to an isomorphism $\Gamma _{\mathfrak{t}}^h\mapsto \Gamma _{\mathfrak{t}}^l$ induced by the ring homomorphism sending $X\mapsto X+{\overline{u_{w_hw_l}}}$ , where ${\overline{u_{w_hw_l}}}=\tfrac{w_h-w_l}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ . Similarly, it restricts to an isomorphism $X_{L_{\mathfrak{t}}^{w_h}}^W\times{\mathbb{P}}_{L_{\mathfrak{t}}^{w_h}}\rightarrow X_{L_{\mathfrak{t}}^{w_l}}^W\times{\mathbb{P}}_{L_{\mathfrak{t}}^{w_l}}$ , where ${\mathbb{P}}_{L_{\mathfrak{t}}^{w_h}}\rightarrow{\mathbb{P}}_{L_{\mathfrak{t}}^{w_l}}$ is the identity and $X_{L_{\mathfrak{t}}^{w_h}}^W\rightarrow X_{L_{\mathfrak{t}}^{w_l}}^W$ is induced by the ring homomorphism sending $X\mapsto X+{\overline{u_{w_hw_l}}}$ .

5.7. Regularity

Let $w_h\in W$ . We want to show that if $\Sigma =\Sigma _C^{\textrm{min}}$ , and $C$ has an almost rational cluster picture and is $y$ -regular, then $\mathring{\mathcal{C}}_{\Delta }^{w_h}$ is a regular scheme.

Lemma 5.19. Consider the model $\mathcal{C}_\Delta ^{w_h}/O_K$ and let $f_h(x)=f(x+w_h)$ . Suppose $\Sigma =\{{\mathfrak{s}}_1,\dots,{\mathfrak{s}}_m\}=\Sigma _C^{\textrm{min}}$ , and $C$ has an almost rational cluster picture and is $y$ -regular. If $P$ is a singular point of $\mathcal{C}_\Delta ^{w_h}$ then

\begin{align*} P\in \textrm{Spec}\,{\dfrac{O_K[X^{\pm 1},Y,Z]}{(\mathcal{F}_M^h(X,Y,Z), X+ u_{hl},Y^{\tilde{m}_{23}}Z^{\tilde{m}_{33}},\pi )}}\subset \mathcal{C}_\Delta ^{w_h}\cap X_M, \end{align*}

for some $l\neq h$ , where $M= M_{L_{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}^{w_h},i}$ for $i=0,\dots,r_{L_{{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l}^{w_h}}$ .

Proof. Denote by $m_\alpha (X)\in O_K[X]$ a lift of the minimal polynomial in $k[X]$ of $\bar \alpha \in \bar{k}$ . By Lemma 5.9, we only need to show that if $P\in \mathcal{C}_\Delta ^{w_h}$ is a singular point then

(8) \begin{equation} P\in \textrm{Spec}\,{\dfrac{O_K[X^{\pm 1},Y,Z]}{(\mathcal{F}_{M_{L,i}}^h(X,Y,Z),m_\alpha (X),Y^{\tilde{m}_{23}}Z^{\tilde{m}_{33}},\pi )}}, \end{equation}

for some $v$ -edge $L=L_{\mathfrak{t}}^{w_h}$ of $\Delta ^{w_h}$ , and some multiple root $\bar{\alpha }$ of $\overline{f_h|_L}$ . We study the polynomial $\mathcal{F}_M^h$ to varying of the matrix $M$ , using [1, §4.5]. Let $g_h(x,y)=y^2-f_h(x)$ . Let $L=L_{\mathfrak{t}}^{w_h}$ and $M=M_{L,i}$ . Note that ${\overline{g_h|_L}}=-{\overline{f_h|_L}}$ . We have $\mathcal{F}_M^h(X,0,Z)={\overline{g_h|_L}}(X)$ for any $i$ . On the other hand, $\mathcal{F}_M^h(X,Y,0)={\overline{g_h|_L}}(X)$ if $i\gt 0$ and $\mathcal{F}_M^h(X,Y,0)={\overline{g_h|_F}}(X,Y)$ if $i=0$ . From the description given in Lemma 5.17, we conclude that for these matrices $M$ the points in (8) are the only possibly singular points of $\mathcal{C}_{\Delta }^{w_h}\cap X_M$ . In particular, this proves that for any $v$ -face $F$ of $\Delta ^{w_h}$ , the points in $X_F$ are non-singular in $\mathcal{C}_{\Delta }^{w_h}$ .

Let $V=V_{\mathfrak{t}}^{w_h}$ or $V=V_0^{w_h}$ and $M=M_{V,j}$ . Since $C$ is $y$ -regular, $p\nmid \deg ({\overline{g_h|_V}})$ by Lemma 4.9. By [1, §4.5] and the fact that the points in $X_F$ are non-singular for all $v$ -faces $F$ , we conclude that $\mathcal{C}_\Delta ^{w_h}$ has no singular point on $X_M$ for these matrices $M$ , as required.

Proposition 5.20. Suppose $\Sigma =\Sigma _C^{\textrm{min}}$ , and $C$ has an almost rational cluster picture and is $y$ -regular, then $\mathcal{C}$ is a regular scheme.

Proof. Lemmas 5.19 and 5.8 show that $\mathring{\mathcal{C}}_{\Delta }^{w_h}$ is regular for every $h$ . Thus their glueing $\mathcal{C}$ is regular as well.

5.8. Separatedness

It remains to prove that $\mathcal{C}$ is a proper scheme. We first show it is separated. Clearly it suffices to prove that $\mathcal{X}/O_K$ is separated. Since the schemes $X_\Delta ^h$ are separated, then the open subschemes $\mathring{X}_\Delta ^{h}$ are separated as well by [9, Proposition 3.3.9]. Consider the open cover $\{V_M^h\}_{h,M}$ of $\mathcal{X}$ . Let $h,l=1,\dots,m$ and let $M_h$ and $M_l$ be matrices associated to proper clusters $\mathfrak{t}_h\in \Sigma _C^{w_h}$ and $\mathfrak{t}_l\in \Sigma _C^{w_l}$ respectively. By [9, Proposition 3.3.6] we want to show

1. (i) $V_{M_h}^h\cap V_{M_l}^l$ is affine,

2. (ii) The canonical homomorphism

   \begin{align*} O_{\mathcal{X}}(V_{M_h}^h)\otimes _{\mathbb{Z}} O_{\mathcal{X}}(V_{M_l}^l)\longrightarrow O_{\mathcal{X}}(V_{M_h}^h\cap V_{M_l}^l) \end{align*}
   is surjective.

The definition of the glueing map (4) implies (i). If $h=l$ , or ${\mathfrak{s}}_l\subseteq \mathfrak{t}_h$ , or ${\mathfrak{s}}_h\subseteq \mathfrak{t}_l$ , then (ii) follows from the separatedness of $\mathring{X}_\Delta ^{h}$ and $\mathring{X}_\Delta ^{l}$ . So assume $l\neq h$ , and $\mathfrak{t}_h,\mathfrak{t}_l\subsetneq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ . Consider the Moebius transformation

\begin{align*} \psi _l:\quad x\mapsto \frac{x}{xw_{hl}^{-1}+1},\quad y\mapsto \frac{y}{(xw_{hl}^{-1}+1)^{g+1}}. \end{align*}

It sends the curve $C^{w_l}$ to the isomorphic hyperelliptic curve

\begin{align*} C_l^h\;:\;y^2=(xw_{hl}^{-1}+1)^{2g+2}f\left ( x(xw_{hl}^{-1}+1)^{-1}+w_l\right ). \end{align*}

As

\begin{align*} f_l^h(x)&\;:\!=\, (xw_{hl}^{-1}+1)^{2g+2}f\left ( x(xw_{hl}^{-1}+1)^{-1}+w_l\right )\\&=c_fw_{hl}^{|\mathfrak{R}|}(xw_{hl}^{-1}+1)^{2g+2-|\mathfrak{R}|}\prod _{r\in \mathfrak{R}\smallsetminus \{w_h\}}\frac{r-w_h}{w_{lh}}\left ( xw_{hl}^{-1}+\frac{r-w_l}{r-w_h}\right ), \end{align*}

every cluster $\mathfrak{s}\in \Sigma _C^{w_l}$ such that $\mathfrak{s}\subsetneq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ , corresponds to a unique cluster ${\mathfrak{s}}^h\in \Sigma _{C_l^h}^0$ of same size, same radius and rational centre $0$ . Moreover,

\begin{align*} \epsilon _{{\mathfrak{s}}^h}=v(c_{f_l^h})+\sum _{r^{\prime}\in{\mathfrak{s}}^h}\rho _{{\mathfrak{s}}^h}+\sum _{r^{\prime}\notin{\mathfrak{s}}^h}v(r')=\epsilon _{\mathfrak{s}}. \end{align*}

Call $\mathfrak{t}_l^h$ the cluster in $\Sigma _{C_l^h}^0$ corresponding to $\mathfrak{t}_l$ . Let $\Delta ^{lh}$ and $\Delta _v^{lh}$ be the Newton polytopes attached to $y^2-f_l^h(x)$ and let $X_\Delta ^{lh}$ be the associated toric scheme (defined in [1, §4.2]). Since $\mathfrak{t}_l\subsetneq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ , the $v$ -faces $F_{\mathfrak{t}_l}$ of $\Delta ^{w_l}$ and $F_{\mathfrak{t}_l^h}$ of $\Delta ^{lh}$ are identical by Lemma 4.3. Furthermore, note that if $\mathfrak{t}_l\lt{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ , then $\rho _{P(\mathfrak{t}_l^h)}\leq \rho _{hl}=\rho _{P(\mathfrak{t}_l)}$ and so $s_2^{V^0}\leq s_2^{V}$ , where $V^0=V_{\mathfrak{t}_l^h}^0$ and $V=V_{\mathfrak{t}_l}^{w_l}$ . Therefore the matrix $M\;:\!=\;M_l$ is also associated to $\mathfrak{t}_l^h$ .

For every $o=1,\dots,m$ , with $o\neq l$ , define

\begin{align*} w_{hlo}=\begin{cases}\dfrac{w_{hl}w_{lo}}{w_{ho}}&\mbox{if } o\neq h,\\ w_{hl}&\mbox{if }o=h,\end{cases} \end{align*}

and write $w_{hlo}=u_{hlo}\pi ^{\rho _{hlo}}$ , where $u_{hlo}\in O_K^\times$ and $\rho _{hlo}\in \mathbb{Z}$ , i.e.

\begin{align*} u_{hlo}=\begin{cases}\dfrac{u_{hl}u_{lo}}{u_{ho}}&\mbox{if } o\neq h,\\u_{hl}&\mbox{if }o=h,\end{cases}\quad \mbox{ and }\quad \rho _{hlo}=\begin{cases}\rho _{hl}+\rho _{lo}-\rho _{ho}&\mbox{if } o\neq h,\\\rho _{hl}&\mbox{if }o=h.\end{cases} \end{align*}

Define

\begin{align*} \tilde{T}_M^{hlo}(X,Y,Z)\;:\!=\;\begin{cases}1+u_{hlo}X^{\rho _{hlo}\tilde{m}_{13}-\tilde{m}_{11}}Y^{\rho _{hlo}\tilde{m}_{23}-\tilde{m}_{21}}Z^{\rho _{hlo}\tilde{m}_{33}-\tilde{m}_{31}}&\mbox{if }\mathfrak{t}_l\supseteq{\mathfrak{s}}_o,\\ u_{hlo}^{-1}X^{\tilde{m}_{11}-\rho _{hlo}\tilde{m}_{13}}Y^{\tilde{m}_{21}-\rho _{hlo}\tilde{m}_{23}}Z^{\tilde{m}_{31}-\rho _{hlo}\tilde{m}_{33}}+1&\mbox{if }\mathfrak{t}_l\not \supseteq{\mathfrak{s}}_o. \end{cases} \end{align*}

We want to show $\tilde{T}_M^{hlo}(X,Y,Z)\in R$ . If $o=h$ then

\begin{align*} \tilde{T}_M^{hlo}(X,Y,Z)=T_M^{hl}(X,Y,Z)\in R. \end{align*}

So assume $o\neq h$ . If ${\mathfrak{s}}_o\subseteq \mathfrak{t}_l$ , then it follows from Lemma 5.4 as ${\mathfrak{s}}_l\wedge{\mathfrak{s}}_o\subsetneq{\mathfrak{s}}_l\wedge{\mathfrak{s}}_h$ and so $\rho _{hlo}=\rho _{lo}$ . On the other hand, if ${\mathfrak{s}}_o\not \subseteq \mathfrak{t}_l$ , then it follows from Lemma 5.5 as $\tilde{m}_{23},\tilde{m}_{33}\gt 0$ and $\rho _{hlo}\leq \max \{\rho _{hl},\rho _{lo}\}$ . Let

\begin{align*} \tilde{T}_M^{hl}(X,Y,Z)\;:\!=\;\prod _{o\neq l}\tilde{T}_M^{hlo}(X,Y,Z). \end{align*}

The Moebius transformation

\begin{align*} K[x^{\pm 1}, y^{\pm 1},\textstyle \prod _{o\neq l}(x+w_{lo})^{-1}]\stackrel{\psi _l}{\longrightarrow } K[x^{\pm 1},y^{\pm 1},\textstyle \prod _{o\neq l}\left ( x+w_{hlo}\right )^{-1}] \end{align*}

considered above induces an isomorphism

\begin{align*} R[T_M^l(X,Y,Z)^{-1}]\xrightarrow{M^{-1}\circ \psi _l\circ M} R[\tilde{T}_M^{hl}(X,Y,Z)^{-1}], \end{align*}

sending

\begin{align*} X&\mapsto X\cdot T_M^{hl}(X,Y,Z)^{-m_{11}-(g+1) m_{21}},\\ Y&\mapsto Y\cdot T_M^{hl}(X,Y,Z)^{-m_{12}-(g+1) m_{22}},\\ Z&\mapsto Z\cdot T_M^{hl}(X,Y,Z)^{-m_{13}-(g+1) m_{23}}. \end{align*}

Then

\begin{align*} \tilde{V}_M^{lh}\;:\!=\;\textrm{Spec}\, R[\tilde{T}_M^{hl}(X,Y,Z)^{-1}] \end{align*}

is an open subscheme of $X_\Delta ^{lh}$ , isomorphic to $V_M^l$ . We can clearly carry out similar constructions for $t_h$ , $M_h$ .

By comparing the Newton polytopes $\Delta _v^{lh}$ and $\Delta _v^{hl}$ , we see that the Moebius transformation $x\mapsto w_{hl}/(w_{lh}^{-1}x)$ , $y\mapsto y/(w_{lh}^{-1}x)^{g+1}$ gives an isomorphism

\begin{align*} \psi \; : \;K[x^{\pm 1},y^{\pm 1}, \prod _{o\neq l}(x+w_{hlo})^{-1}]\longrightarrow K[x^{\pm 1},y^{\pm 1}, \prod _{o\neq h}(x+w_{lho})^{-1}] \end{align*}

which induces a birational map $X_\Delta ^{hl}\dashrightarrow X_\Delta ^{lh}$ , defined on the open set $\tilde{V}_{M_h}^{hl}$ of $X_\Delta ^{hl}$ . In particular, there exists an open set $\tilde{V}_{M_h}^{lh}$ of $X_\Delta ^{lh}$ , isomorphic to $V_{M_h}^h$ via the map induced by $\psi _h^{-1}\circ \psi$ .

Recall the definition of $\phi$ in (1), which induces the glueing map between $V_{M_l}^l$ and $V_{M_h}^h$ . Since the following diagram

is commutative, then the surjectivity of

\begin{align*} O_{\mathcal{X}}(V_{M_h}^h)\otimes _{\mathbb{Z}} O_{\mathcal{X}}(V_{M_l}^l)\longrightarrow O_{\mathcal{X}}(V_{M_h}^h\cap V_{M_l}^l) \end{align*}

follows from the separatedness of $X_\Delta ^{lh}$ .

5.9. Properness

By [2, IV.15.7.10], it remains to show that $\mathcal{C}_s$ is proper. From [9, Exercise 3.3.11], we only need to prove that the $1$ -dimensional subscheme $\Gamma _{\mathfrak{t}}$ is proper for every $\mathfrak{t}={\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ . Indeed every other component is entirely contained in a model $\mathcal{C}_\Delta ^{w_h}$ , which is proper (see Section 5.5). Let $\mathfrak{t}={\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ for some $h,l=1,\dots,m$ , with $h\neq l$ . For any $o=1,\dots, m$ such that ${\mathfrak{s}}_o\subset \mathfrak{t}$ , let $\mathfrak{t}_o$ be the unique child of $\mathfrak{t}$ with ${\mathfrak{s}}_o\subseteq \mathfrak{t}_o\lt \mathfrak{t}$ . Then $\Gamma _{\mathfrak{t}}$ is equal to the glueing of the schemes

\begin{align*} \textrm{Spec}\,\frac{R[T_M^{o}(X,Y,Z)^{-1}]}{\left (\mathcal{F}_M^o(X,Y,Z),Z,\pi \right )},\quad M=M_{L_{\mathfrak{t}}^{w_o},0},M_{V_{\mathfrak{t}}^{w_o},0}, \end{align*}

and

\begin{align*} \textrm{Spec}\,\frac{R[T_M^{o}(X,Y,Z)^{-1}]}{\left (\mathcal{F}_M^o(X,Y,Z),Y,\pi \right )},\quad M=M_{V_{\mathfrak{t}_o}^{w_o},r_{V_{\mathfrak{t}_o}^{w_o}}}, \end{align*}

for all $o$ such that ${\mathfrak{s}}_o\subset \mathfrak{t}$ , through the isomorphism (4) and the glueing maps in the definition of $\mathcal{C}_\Delta ^{w_o}$ . In particular, for any $o$ as above there exists a natural birational map $s_o\;:\;\Gamma _{\mathfrak{t}}\dashrightarrow \bar{X}_{F_{\mathfrak{t}}^{w_o}}$ which is defined as the identity morphism on the dense open $ X_{F_{\mathfrak{t}}^{w_o}}=\Gamma _{\mathfrak{t}}\cap\mathring{\mathcal{C}}_\Delta ^{w_o}$ .

Let $D/k$ be a normal curve, let $P\in D$ and let $D\smallsetminus \{P\}\stackrel{g}{\longrightarrow } \Gamma _{\mathfrak{t}}$ be a non-constant morphism of curves. We want to show that $g$ extends to $D$ . For every $o$ as above, $\bar{X}_{F_{\mathfrak{t}}^{w_o}}$ is proper, so the birational map

\begin{align*} g_o\;:\!=\;s_o\circ g\;:\;D\smallsetminus \{P\}\dashrightarrow \bar{X}_{F_{\mathfrak{t}}^{w_o}} \end{align*}

extends to a morphism $\bar{g}_o\; : \;D\longrightarrow \bar{X}_{F_{\mathfrak{t}}^{w_o}}$ . If

\begin{align*} P_o\;:\!=\;\bar{g}_o(P)\in \left (\bar{X}_{F_{\mathfrak{t}}^{w_o}}\cap\mathring{\mathcal{C}}_\Delta ^{w_o}\right )= s_o\left (\Gamma _{\mathfrak{t}}\cap\mathring{\mathcal{C}}_\Delta ^{w_o}\right ) \end{align*}

for some $o$ such that ${\mathfrak{s}}_o\subset \mathfrak{t}$ (we will later show this is always the case), then there exists an open neighbourhood $U$ of $P_o$ such that $U\subseteq \left (\bar{X}_{F_{\mathfrak{t}}^{w_o}}\cap\mathring{\mathcal{C}}_\Delta ^{w_o}\right )$ and so $s_o|_{s_o^{-1}(U)}^U$ is an isomorphism. Since $P\in \bar{g}_o^{-1}(U)$ , the map

\begin{align*} \bar{g}_o^{-1}(U)\xrightarrow{\bar{g}_o|_{\bar{g}_o^{-1}(U)}^U}U\xrightarrow{\big ( s_o|_{s_o^{-1}(U)}^U\big )^{-1}}s_o^{-1}(U)\hookrightarrow \Gamma _{\mathfrak{t}}, \end{align*}

induces an extension $D\longrightarrow \Gamma _{\mathfrak{t}}$ of $g$ .

Suppose that $P_o\notin \bar{X}_{F_{\mathfrak{t}}^{w_o}}\cap\mathring{\mathcal{C}}_\Delta ^{w_o}$ for any $o$ such that ${\mathfrak{s}}_o\subset \mathfrak{t}$ . From Section 5.5 we have

(9) \begin{equation} P_o\in S_M=\textrm{Spec}\,{\dfrac{R}{(\mathcal{F}_M^o(X,Y,Z),\prod _l\left ( X+ u_{ol}\right ),Z,\pi )}}, \end{equation}

where $M=M_{L_{\mathfrak{t}}^{w_o},0}$ , and the product runs over all $l\neq o$ such that $\mathfrak{t}={\mathfrak{s}}_o\wedge{\mathfrak{s}}_l$ . In particular $P_o$ is a point of each irreducible component of $\bar{X}_{F_{\mathfrak{t}}^{w_o}}$ by Lemma 5.17. Let $h\neq o$ such that $X+u_{oh}$ vanishes at $P_o$ . Let $\xi$ be the generic point of $D$ and let $\xi _o=g_o(\xi )$ , $\xi _h=g_h(\xi )$ be generic points of $\bar{X}_{F_{\mathfrak{t}}^{w_o}}$ and $\bar{X}_{F_{\mathfrak{t}}^{w_h}}$ respectively. Then the birational maps $s_o$ and $s_h$ give

where we denote by $\phi _{g_o}$ and $\phi _{g_h}$ the homomorphisms between function fields induced by $g_o$ and $g_h$ . The vertical isomorphism is induced by the map

\begin{align*} \frac{R[T_M^o(X,Y,Z)^{-1}]}{\left ( \mathcal{F}_M^o(X,Y,Z),Z\right )}\longrightarrow \frac{R[T_M^h(X,Y,Z)^{-1}]}{\left ( \mathcal{F}_M^h(X,Y,Z),Z\right )} \end{align*}

which sends (see Section 5.3 and Lemma 5.7)

\begin{align*} X+u_{oh}\mapsto X\cdot T_M^{ho}(X,Y,Z)^{m_{11}}+u_{oh}=X\left (1+u_{ho}X^{-1}\right )+u_{oh}=X. \end{align*}

But the rational function $X+u_{oh}$ vanishes at $P_o$ , while $X$ does not vanish at $P_h$ by (9). This gives a contradiction, as $\bar{g}_o(P)=P_o$ and $\bar{g}_h(P)=P_h$ .

5.10. Genus

Suppose $\Sigma =\{{\mathfrak{s}}_1,\dots,{\mathfrak{s}}_m\}=\Sigma _C^{\textrm{min}}$ , and $C$ has an almost rational cluster picture and is $y$ -regular. In the previous subsections we proved that $\mathcal{C}/O_K$ is a proper regular model of $C$ . Let $\mathfrak{t}\in \Sigma _C^{w_h}$ be a proper cluster.

Proposition 5.21. Let $\mathfrak{t}\in \Sigma _C^{w_h}$ . Then $\Gamma _{\mathfrak{t}}$ is isomorphic to the smooth projective $1$ -dimensional scheme given by

\begin{align*} Y^{D_{\mathfrak{t}}}=\prod _{\mathfrak{s}\in \tilde{\mathfrak{t}}^W}(X-{\overline{u_{w_{\mathfrak{s}} w_h}}}){\overline{f_{\mathfrak{t},w_h}}}(X) \end{align*}

where ${\overline{u_{w_{\mathfrak{s}} w_h}}}=\frac{w_{\mathfrak{s}}-w_h}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ , and $w_{\mathfrak{s}}$ is any rational centre of $\mathfrak{s}$ .

In particular,

1. (1) if $D_{\mathfrak{t}}=1$ , then $\Gamma _{\mathfrak{t}}\simeq{\mathbb{P}}^1_k$ ;

2. (2) if $D_{\mathfrak{t}}=2$ and $\mathfrak{t}$ is übereven, then $\Gamma _{\mathfrak{t}}$ is the disjoint union of two ${\mathbb{P}}^1$ s over some quadratic extension of $k$ ;

3. (3) in all other cases, $\Gamma _{\mathfrak{t}}$ is a hyperelliptic curve of genus $g(\mathfrak{t})$ .

Proof. The first part of the proposition follows from Proposition 5.18.

For the second part of the statement note that if $D_{\mathfrak{t}}=1$ then the result follows. Suppose $D_{\mathfrak{t}}=2$ . Then $p\neq 2$ as $C$ is $y$ -regular. Note that since $\Sigma =\Sigma _C^{\textrm{min}}$ , the proper clusters in $\Sigma ^W$ correspond to the proper clusters in $\Sigma _C^{\textrm{rat}}$ . Recall the definition of $\tilde{\mathfrak{t}}$ given in Definition 4.13. Let $h(X)=\prod _{\mathfrak{s}\in \tilde{\mathfrak{t}}^W}(X-{\overline{u_{w_{\mathfrak{s}} w_h}}}){\overline{f_{\mathfrak{t},w_h}}}(X)$ .

Suppose $\mathfrak{t}$ is übereven. Then all its children are (proper) rational cluster by Lemma 3.30 since they are even and $p\neq 2$ . In particular $b_{\mathfrak{t}}=1$ by Lemma 3.18 and so $\epsilon _{\mathfrak{t}}\in 2\mathbb{Z}$ and $\tilde{\mathfrak{t}}=\tilde{\mathfrak{t}}^W=\varnothing$ since it equals the set of odd rational children. Moreover, $\mathfrak{t}=\bigcup _{\mathfrak{s}\lt \mathfrak{t},\,\mathfrak{s}\,\text{proper}}\mathfrak{s}$ , and so ${\overline{f_{\mathfrak{t},w_h}}}\in k$ . Thus $h(X)\in k$ .

Now suppose $h(X)\in k$ . Then $\tilde{\mathfrak{t}}^W=\varnothing$ and $\mathfrak{t}=\bigcup _{\mathfrak{s}\lt \mathfrak{t}}\mathfrak{s}$ , where $\mathfrak{s}$ runs through all children $\mathfrak{s}\in \Sigma ^W$ of $\mathfrak{t}$ . The non-proper clusters in $\Sigma ^W$ are of the form $\{w_l\}$ for some $l=1,\dots,m$ . If $\{w_l\}\lt \mathfrak{t}$ , then $\mathfrak{t}={\mathfrak{s}}_l$ , but in that case $\mathfrak{t}$ would not equal the union of its children in $\Sigma ^W$ . Hence $\mathfrak{t}$ has no non-proper children. It follows that $\tilde{\mathfrak{t}}=\tilde{\mathfrak{t}}^W$ and $\mathfrak{t}$ equals the union of its proper rational children. In particular, $\mathfrak{t}$ has two or more children in $\Sigma _C^{\textrm{rat}}$ , so $b_{\mathfrak{t}}=1$ , by Lemma 3.18. But then $\tilde{\mathfrak{t}}$ is the set of odd children of $\mathfrak{t}$ as $\epsilon _{\mathfrak{t}}\in 2\mathbb{Z}$ , and so all rational children of $\mathfrak{t}$ are even.

It only remains to prove that if $h(x)\notin k$ , then the genus of $\Gamma _{\mathfrak{t}}$ is $g(\mathfrak{t})$ . Since $h(X)$ is a separable polynomial, we need to show that

\begin{align*} \deg h=\frac{|\mathfrak{t}|-\sum _{\mathfrak{s}\in \Sigma _C^{\textrm{rat}},\,\mathfrak{s}\lt \mathfrak{t}}|\mathfrak{s}|}{b_{\mathfrak{t}}}+\left |\tilde{\mathfrak{t}}\right |. \end{align*}

It suffices to prove that if $\mathfrak{s}\in \Sigma _C^{\textrm{rat}}$ is a non-proper rational child of $\mathfrak{t}$ different from $\{w_h\}$ , then $b_{\mathfrak{t}}=1$ and $\mathfrak{s}\in \tilde{\mathfrak{t}}$ . Suppose $\mathfrak{s}=\{r\}$ is such a rational cluster. Since $r\in \mathfrak{t}$ , we have $v(r-w_h)\geq \rho _{\mathfrak{t}}$ . Suppose $v(r-w_h)\gt \rho _{\mathfrak{t}}$ . Then $\mathfrak{s}\in \Sigma _C^{w_h}$ , as $\mathfrak{s}\lt \mathfrak{t}$ and $r\neq w_h$ . But this contradicts our choice of $W$ . Then $\rho _{\mathfrak{t}}=v(r-w_h)\in \mathbb{Z}$ and so $b_{\mathfrak{t}}=1$ . It follows that $\tilde{\mathfrak{t}}$ is the set of odd children of $\mathfrak{t}$ . Thus $\mathfrak{s}\in \tilde{\mathfrak{t}}$ .

5.11. Minimal regular NC model

Suppose the base extended curve $C_{K^{nr}}$ is $y$ -regular and has an almost rational cluster picture. Consider the model $\mathcal{C}/O_{K^{nr}}$ constructed before with $\Sigma =\Sigma _{C_{K^{nr}}}^{\textrm{min}}$ . We want to see what components of $\mathcal{C}_s$ should be blown down to obtain the minimal regular model with normal crossings. Recall [1, §5]. Let $\Sigma _{K^{nr}}=\Sigma _{C_{K^{nr}}}^{\textrm{rat}}$ and fix a proper cluster $\mathfrak{t}\in \Sigma _{C_{K^{nr}}}^{w_h}$ .

Suppose first $\mathfrak{t}\neq{\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ for all $l=1,\dots,m$ with $l\neq h$ . Equivalently, $\mathfrak{t}$ has at most one proper child in $\Sigma _{K^{nr}}$ . Then $\Gamma _{\mathfrak{t}}\simeq \bar{X}_{F_{\mathfrak{t}}^{w_h}}$ and can be seen entirely in $\mathring{\mathcal{C}}_{\Delta }^{w_h}$ . In particular, if $\Gamma _{\mathfrak{t}}$ can be blown down then $F_{\mathfrak{t}}^{w_h}$ is a removable or contractible $v$ -face (see [1, Theorem 5.7]). By Lemma 4.3, we find

• $F_{\mathfrak{t}}^{w_h}$ is removable if and only if $\mathfrak{t}=\mathfrak{R}$ with a child in $\Sigma _{K^{nr}}$ of size $2g+1$ .

• $F_{\mathfrak{t}}^{w_h}$ is contractible if and only if either $|\mathfrak{t}|=2$ and $\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}\in \mathbb{Z}$ or $\mathfrak{t}$ has a proper rational child $\mathfrak{s}\in \Sigma _{K^{nr}}$ , of size $2g$ , and $\frac{\epsilon _{\mathfrak{t}}}{2}-g\rho _{\mathfrak{t}}\in \mathbb{Z}$ .

Recall Definition 4.20. Note that $F_{\mathfrak{t}}^{w_h}$ is removable if and only if $\mathfrak{t}$ is removable. In this case, $F_{\mathfrak{t}}^{w_h}$ can be ignored for the construction of $\mathcal{C}_\Delta ^{w_h}$ (for any $h$ since $\mathfrak{t}=\mathfrak{R}$ ), and so $\mathfrak{t}$ can be ignored for the construction of $\mathcal{C}$ .

Assume now $F_{\mathfrak{t}}^{w_h}$ contractible. We want to understand when $\Gamma _{\mathfrak{t}}$ can be blown down. First consider the case $|\mathfrak{t}|=2$ and $\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}\in \mathbb{Z}$ . Then $\Gamma _{\mathfrak{t}}$ intersects other components of $\mathcal{C}_s$ in $2$ points (as $V_{\mathfrak{t}}^{w_h}$ gives two chains of ${\mathbb{P}}^1$ s and the $v$ -edges $V_0^{w_h}$ and $L_{\mathfrak{t}}^{w_h}$ give no component in $\mathcal{C}_{\Delta,s}^{w_h}$ ). To have self-intersection $-1$ , $\Gamma _{\mathfrak{t}}$ has to have multiplicity $\gt 1$ . It follows from Lemma 5.16 that $\rho _{\mathfrak{t}}\notin \mathbb{Z}$ , as $\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}\in \mathbb{Z}$ . Moreover, by Lemma 3.12, one has $\rho _{\mathfrak{t}}\in \frac{1}{2}\mathbb{Z}$ . Therefore $\epsilon _{\mathfrak{t}}$ is odd and the multiplicity of $\Gamma _{\mathfrak{t}}$ is $2$ . Let $r\;:\!=\;r_{V_{\mathfrak{t}}^{w_h}}$ and consider

\begin{align*} \gamma _{\mathfrak{t}} s_{\mathfrak{t}}=\frac{n_0}{d_0}\gt \frac{n_1}{d_1}\gt \dots \gt \frac{n_r}{d_r}\gt \frac{n_{r+1}}{d_{r+1}}=\gamma _{\mathfrak{t}}\left ( s_{\mathfrak{t}}-\rho _{\mathfrak{t}}+\rho _{P(\mathfrak{t})}\right ) \end{align*}

given by $V_{\mathfrak{t}}^{w_h}$ . If $\Gamma _{\mathfrak{t}}$ can be blown down then $d_1=1$ . Since $\gamma _{\mathfrak{t}} s_{\mathfrak{t}}=-\frac{\epsilon _{\mathfrak{t}}}{2}+2\rho _{\mathfrak{t}}$ , we have $d_0=2$ . In particular $d_1=1$ if and only if $\rho _{\mathfrak{t}}-\rho _{P(\mathfrak{t})}=\frac{n_0}{d_0}-\frac{n_{r+1}}{d_{r+1}}\geq \frac{1}{2}$ (see also [1, Remark 3.15]). Thus if $|\mathfrak{t}|=2$ , then $\Gamma _{\mathfrak{t}}$ can be blown down if and only if $\rho _{\mathfrak{t}}\notin \mathbb{Z}$ , $\epsilon _{\mathfrak{t}}$ odd, $\rho _{P(\mathfrak{t})}\leq \rho _{\mathfrak{t}}-\frac{1}{2}$ . Note that this is case (1) of Definition 4.20.

Second consider the case $|\mathfrak{t}|=2g+2$ with a proper rational child $\mathfrak{s}$ of size $2g$ and $\tfrac{\epsilon _{\mathfrak{t}}}{2}-g\rho _{\mathfrak{t}}\in \mathbb{Z}$ . The argument is very similar to the previous one. If $\Gamma _{\mathfrak{t}}$ can be blown down then it must have multiplicity $\gt 1$ and this implies $\rho _{\mathfrak{t}}\notin \mathbb{Z}$ again by Lemma 5.16. From Lemma 3.12 it follows that $(|\mathfrak{t}|-|\mathfrak{s}|)\rho _{\mathfrak{t}}\in \mathbb{Z}$ , so $\rho _{\mathfrak{t}}\in \frac{1}{2}\mathbb{Z}$ . Then $m_{\mathfrak{t}}=2$ and

\begin{align*} \frac{v(c_f)}{2}=\frac{\epsilon _{\mathfrak{t}}}{2}-(g+1)\rho _{\mathfrak{t}}\in \tfrac{1}{2}\mathbb{Z}\setminus \mathbb{Z}, \end{align*}

so $v(c_f)$ odd. Let $r\;:\!=\;r_{V_{\mathfrak{s}}^{w_h}}$ and consider

\begin{align*} \gamma _{\mathfrak{s}}s_{\mathfrak{s}}=\frac{n_0}{d_0}\gt \frac{n_1}{d_1}\gt \dots \gt \frac{n_r}{d_r}\gt \frac{n_{r+1}}{d_{r+1}}=\gamma _{\mathfrak{s}}(s_{\mathfrak{s}}-\rho _{\mathfrak{s}}+\rho _{\mathfrak{t}}) \end{align*}

given by $V_{\mathfrak{s}}^{w_h}$ . If $\Gamma _{\mathfrak{t}}$ can be blown down then $d_r=1$ . Recall that $\epsilon _{\mathfrak{s}}-|\mathfrak{s}|\rho _{\mathfrak{s}}=\epsilon _{\mathfrak{t}}-|\mathfrak{s}|\rho _{\mathfrak{t}}$ . Then $\gamma _{\mathfrak{s}}(s_{\mathfrak{s}}-\rho _{\mathfrak{s}}+\rho _{\mathfrak{t}})=-\frac{\epsilon _{\mathfrak{t}}}{2}+(g+1)\rho _{\mathfrak{t}}$ , so $d_{r+1}=2$ . In particular $d_r=1$ if and only if $\rho _{\mathfrak{s}}-\rho _{\mathfrak{t}}=\frac{n_0}{d_0}-\frac{n_{r+1}}{d_{r+1}}\geq \frac{1}{2}$ . Thus if $\mathfrak{t}$ has size $2g+2$ and has a unique proper rational child $\mathfrak{s}\in \Sigma _{K^{nr}}$ , then $\Gamma _{\mathfrak{t}}$ can be blown down if and only if $|\mathfrak{s}|=2g$ , $\rho _{\mathfrak{t}}\notin \mathbb{Z}$ , $v(c_f)$ odd, $\rho _{\mathfrak{s}}\geq \rho _{\mathfrak{t}}+\frac{1}{2}$ . This is case (2) of Definition 4.20.

Finally, if $|\mathfrak{t}|=2g+1$ , $\mathfrak{t}$ has a proper child $\mathfrak{s}\in \Sigma _{K^{nr}}$ of size $2g$ and $\frac{\epsilon _{\mathfrak{t}}}{2}-g\rho _{\mathfrak{t}}\in \mathbb{Z}$ , then $\rho _{\mathfrak{t}}\in \mathbb{Z}$ , as $(|\mathfrak{t}|-|\mathfrak{s}|)\rho _{\mathfrak{t}}\in \mathbb{Z}$ . It follows that $\epsilon _{\mathfrak{t}}\in \mathbb{Z}$ and so $m_{\mathfrak{t}}=1$ . This implies the self-intersection of $\Gamma _{\mathfrak{t}}$ is not $-1$ , since it intersects the rest of $\mathcal{C}_{\mathfrak{t}}$ in at least two points as before. Hence in this case $\Gamma _{\mathfrak{t}}$ can never be blown down.

Now assume there exists $l\neq h$ such that $\mathfrak{t}={\mathfrak{s}}_h\wedge{\mathfrak{s}}_l$ . Then $\mathfrak{t}$ is not minimal. Let $\mathfrak{t}_h,\mathfrak{t}_l\in \Sigma _{K^{nr}}$ be such that ${\mathfrak{s}}_h\subseteq \mathfrak{t}_h\lt \mathfrak{t}$ and ${\mathfrak{s}}_l\subseteq \mathfrak{t}_l\lt \mathfrak{t}$ . Suppose $\Gamma _{\mathfrak{t}}$ irreducible. If $|\mathfrak{t}|\leq 2g$ (or, equivalently, $\mathfrak{t}$ is not the largest non-removable cluster), then $\Gamma _{\mathfrak{t}}$ intersects at least other $3$ components of $\mathcal{C}_s$ (given by $\mathfrak{t}_h,\mathfrak{t}_l$ , and $P(\mathfrak{t})$ ). So it cannot be contracted to obtain a model with normal crossings. A similar argument holds if there exists $o\neq l$ such that ${\mathfrak{s}}_o\wedge{\mathfrak{s}}_h=\mathfrak{t}$ : at least $3$ components (given by $\mathfrak{t}_h$ , $\mathfrak{t}_l$ and $\mathfrak{t}_o$ ) intersect $\Gamma _{\mathfrak{t}}$ , so blowing down $\Gamma _{\mathfrak{t}}$ would make the model lose normal crossings. Assume then $|\mathfrak{t}|\gt 2g$ and ${\mathfrak{s}}_o\wedge{\mathfrak{s}}_h\neq \mathfrak{t}$ for all $o\neq l$ . Then $\Gamma _{\mathfrak{t}}$ intersects at least other $2$ components of $\mathcal{C}_s$ given by $V_{\mathfrak{t}_h}^{w_h}$ and $V_{\mathfrak{t}_l}^{w_l}$ . Firstly, if $\Gamma _{\mathfrak{t}}$ can be blown down, then $m_{\mathfrak{t}}\gt 1$ . But $\rho _{\mathfrak{t}}=\rho _{hl}\in \mathbb{Z}$ . Then $m_{\mathfrak{t}}$ is at most $2$ . If $m_{\mathfrak{t}}=2$ then $D_{\mathfrak{t}}=1$ , that implies $\epsilon _{\mathfrak{t}}$ odd and $\Gamma _{\mathfrak{t}}\simeq{\mathbb{P}}^1$ by Proposition 5.21. It also follows $s_{\mathfrak{t}}\in \tfrac{1}{2}\mathbb{Z}\setminus \mathbb{Z}$ . If $\mathfrak{t}$ is odd then this implies that $V_{\mathfrak{t}}^{w_h}$ gives a ${\mathbb{P}}^1$ intersecting $\Gamma _{\mathfrak{t}}$ . Since that would be a third component intersecting $\Gamma _{\mathfrak{t}}$ , the cluster $\mathfrak{t}$ has to be even. Hence $\mathfrak{t}=\mathfrak{R}$ and $|\mathfrak{t}|=2g+2$ . Then $\epsilon _{\mathfrak{t}}$ is odd if and only if $v(c_f)$ is odd, as $\rho _{\mathfrak{t}}\in \mathbb{Z}$ . Now, $L_{\mathfrak{t}}^{w_h}$ gives some ${\mathbb{P}}^1$ s intersecting $\bar{X}_{F_{\mathfrak{t}}^{w_h}}\subset \mathcal{C}_{\Delta,s}^{w_h}$ . All these ${\mathbb{P}}^1$ s are not in $\mathring{\mathcal{C}}_{\Delta,s}^{w_h}$ (and so in $\mathcal{C}_s$ ) if and only if $\mathfrak{t}_h\cup \mathfrak{t}_l=\mathfrak{t}$ . In particular, $\mathfrak{t}_h$ and $\mathfrak{t}_l$ are either both even or both odd. If $\mathfrak{t}_h$ is even, then $\gamma _{\mathfrak{t}_h}=2$ , and so the component given by $V_{\mathfrak{t}_h}^{w_h}$ has multiplicity at least $2$ . The self-intersection of $\Gamma _{\mathfrak{t}}$ could not be $-1$ in this case. Assume $\mathfrak{t}_h$ is odd. Let $r\;:\!=\;r_{V_{\mathfrak{t}_h}^{w_h}}$ and consider

\begin{align*} \gamma _{\mathfrak{t}_h}s_{\mathfrak{t}_h}=\frac{n_0}{d_0}\gt \frac{n_1}{d_1}\gt \dots \gt \frac{n_r}{d_r}\gt \frac{n_{r+1}}{d_{r+1}}=\gamma _{\mathfrak{t}_h}\left ( s_{\mathfrak{t}_h}-\frac{\rho _{\mathfrak{t}_h}-\rho _{\mathfrak{t}}}{2}\right ) \end{align*}

given by $V_{\mathfrak{t}_h}^{w_h}$ . We want $d_r=1$ . Since

\begin{align*} \gamma _{\mathfrak{t}_h}\left ( s_{\mathfrak{t}_h}-\frac{\rho _{\mathfrak{t}_h}-\rho _{\mathfrak{t}}}{2}\right )=-\frac{\epsilon _{\mathfrak{t}}}{2}+\frac{|\mathfrak{t}_h|-1}{2}\rho _{\mathfrak{t}}\in \tfrac{1}{2}\mathbb{Z}\smallsetminus \mathbb{Z}, \end{align*}

we have $d_{r+1}=2$ . As before $d_r=1$ if and only if $\frac{\rho _{\mathfrak{t}_h}-\rho _{\mathfrak{t}}}{2}=\frac{n_0}{d_0}-\frac{n_{r+1}}{d_{r+1}}\geq \frac{1}{2}$ and similarly for $\mathfrak{t}_l$ . Thus if $\mathfrak{t}$ has two or more rational children and $\Gamma _{\mathfrak{t}}$ is irreducible then it can be blown down if and only if $v(c_f)$ is odd and $\mathfrak{t}=\mathfrak{R}$ is union of its $2$ odd rational children $\mathfrak{t}_h$ and $\mathfrak{t}_l$ , satisfying $\rho _{\mathfrak{t}_h}\geq \rho _{\mathfrak{t}}+1$ , $\rho _{\mathfrak{t}_l}\geq \rho _{\mathfrak{t}}+1$ . This is case (3) of Definition 4.20.

Suppose now $\Gamma _{\mathfrak{t}}$ reducible. By Proposition 5.21 the cluster $\mathfrak{t}$ is übereven, $\epsilon _{\mathfrak{t}}$ is even and $\Gamma _{\mathfrak{t}}$ is the disjoint union of $\Gamma _{\mathfrak{t}}^-\simeq{\mathbb{P}}^1$ and $\Gamma _{\mathfrak{t}}^+\simeq{\mathbb{P}}^1$ . As before, both $\Gamma _{\mathfrak{t}}^-$ and $\Gamma _{\mathfrak{t}}^+$ intersect at least other two components (given by the proper children of $\mathfrak{t}$ ). But then neither $\Gamma _{\mathfrak{t}}^-$ nor $\Gamma _{\mathfrak{t}}^+$ has self-intersection $-1$ , as $m_{\mathfrak{t}}=1$ .

We have showed that, for a rational cluster $\mathfrak{t}\in \Sigma _{K^{nr}}$ , an irreducible component of $\Gamma _{\mathfrak{t}}$ can be blown down if and only if $\mathfrak{t}$ is contractible. Moreover, in this case, $\Gamma _{\mathfrak{t}}$ is irreducible. It remains to show that after blowing down all components $\Gamma _{\mathfrak{t}}$ where $\mathfrak{t}$ is a contractible cluster, no other component can be blown down. First note that if $\mathfrak{t}$ is a contractible cluster, then $m_{\mathfrak{t}}=2$ and $\Gamma _{\mathfrak{t}}$ intersects one or two other components of multiplicity $1$ at two points in total. If it intersects only one component, then after the blowing down, the latter will have a node and will not be isomorphic to ${\mathbb{P}}^1$ . If $\Gamma _{\mathfrak{t}}$ intersects two components and those intersect something else in $\mathcal{C}_s$ , then they will not have self-intersection $-1$ also when $\Gamma _{\mathfrak{t}}$ is blown down. Therefore suppose that one of those two does not intersect any other component of $\mathcal{C}_s$ . If we are in case (1) or case (2), it is easy to see that this never happens. Indeed, in those cases, $\Gamma _{\mathfrak{t}}$ intersects non-open-ended chains of ${\mathbb{P}}^1$ s. Then without loss of generality assume to be in case (3) and that $\Gamma _{\mathfrak{t}_h}$ is the component that can be blown down once $\Gamma _{\mathfrak{t}}$ has been contracted. This implies ${\mathfrak{s}}_h=\mathfrak{t}_h$ and $\rho _{{\mathfrak{s}}_h}=\rho _{\mathfrak{t}}+1$ . Then $b_{{\mathfrak{s}}_h}=1$ and $\epsilon _{{\mathfrak{s}}_h}=\epsilon _{\mathfrak{t}}+|{\mathfrak{s}}_h|$ . Since both $\epsilon _{\mathfrak{t}}$ and ${\mathfrak{s}}_h$ are odd, we have $\epsilon _{{\mathfrak{s}}_h}\in 2\mathbb{Z}$ . So $D_{{\mathfrak{s}}_h}=2$ and $\tilde{\mathfrak{s}}_h$ is the set of rational children of ${\mathfrak{s}}_h$ . Hence $g({\mathfrak{s}}_h)=\left \lfloor \frac{|{\mathfrak{s}}_h|-1}{2}\right \rfloor \geq 1$ since $|{\mathfrak{s}}_h|\geq 3$ . But then $\Gamma _{{\mathfrak{s}}_h}$ cannot be blown down.

5.12. Galois action

Consider the base extended hyperelliptic curve $C_{K^{nr}}/K^{nr}$ . The rational clusters of $C_{K^{nr}}$ and their corresponding rational centres are then over $K^{nr}$ . Denote $\Sigma _{K^{nr}}=\Sigma _{C_{K^{nr}}}^{\textrm{rat}}$ . For any proper cluster $\mathfrak{s}\in \Sigma _{K^{nr}}$ , let $G_{\mathfrak{s}}=\textrm{Stab}_{G_K}(\mathfrak{s})$ , $K_{\mathfrak{s}}=\left ( K^{\textrm{s}}\right )^{G_{\mathfrak{s}}}$ and $k_{\mathfrak{s}}$ be the residue field of $K_{\mathfrak{s}}$ . Let $\Sigma _{C_{K^{nr}}}^{\textrm{min}}=\{{\mathfrak{s}}_1,\dots,{\mathfrak{s}}_m\}$ be the set of rationally minimal clusters of $C_{K^{nr}}$ . Fix a set $W=\{w_1,\dots,w_m\}\subset K^{nr}$ of corresponding rational centres. By Lemma A.1, we can assume this choice to be $G_K$ -equivariant, that is for any $\sigma \in G_K$ , one has $\sigma (w_l)=w_h$ if and only if $\sigma ({\mathfrak{s}}_l)={\mathfrak{s}}_h$ . We can also require that $w_h\in{\mathfrak{s}}_h$ if ${\mathfrak{s}}_h\cap K_{{\mathfrak{s}}_h}\neq \varnothing$ . Similarly, for any proper cluster $\mathfrak{t}\in \Sigma _{K^{nr}}\setminus \Sigma _{C_{K^{nr}}}^{\textrm{min}}$ , fix a rational centre $w_{\mathfrak{t}}$ in such a way that $w_{\sigma (t)}=\sigma (w_{\mathfrak{t}})$ for any $\sigma \in G_K$ . Set $w_{{\mathfrak{s}}_o}\;:\!=\;w_o$ for any $o=1,\dots,m$ .

Lemma 5.22. With the choices above, for any $h=1,\dots,m$ , the set of proper clusters in $\Sigma _{C_{K^{nr}}}^{w_h}$ coincides with $\mathring{\Sigma }_{C_{K^{nr}}}^{w_h}$ .

Proof. Suppose by contradiction that there exists a non-proper cluster $\{r\}=\mathfrak{s}\in \Sigma _{C_{K^{nr}}}^{w_h}$ , with $r\neq w_h$ . Note that $r\in{\mathfrak{s}}_h$ and so $\mathfrak{s}\lt{\mathfrak{s}}_h$ . Recall that since $\mathfrak{s}$ is a cluster centred at $w_h$ , it is cut out by the disc $\mathcal{D}=\{x\in \bar{K}\mid v(x-w_h)\geq \rho _{\mathfrak{s}}^{w_h}\}$ , with $\rho _{\mathfrak{s}}^{w_h}=v(r-w_h)\gt \rho _{{\mathfrak{s}}_h}$ . This implies that $w_h\notin \mathfrak{R}$ , otherwise $w_h\in \mathfrak{s}$ and $|\mathfrak{s}|\geq 2$ . In particular, $w_h\notin{\mathfrak{s}}_h$ . For our choice of $w_h$ , it follows that ${\mathfrak{s}}_h\cap K_{{\mathfrak{s}}_h}=\varnothing$ . Therefore $r\notin K_{{\mathfrak{s}}_h}$ and so there exists $\sigma \in G_{{\mathfrak{s}}_h}$ such that $\sigma (r)\neq r$ . Since $w_h\in K_{{\mathfrak{s}}_h}$ we have

\begin{align*} v(\sigma (r)-w_h)=v(\sigma (r-w_h))=v(r-w_h)=\rho _{\mathfrak{s}}^{w_h}. \end{align*}

But then $\sigma (r)\in \mathfrak{s}$ , and so $|\mathfrak{s}|\geq 2$ , a contradiction.

Assume that $C_{K^{nr}}$ is $y$ -regular and has an almost rational cluster picture. By the previous lemma, from the set of rational centres $W$ we can construct the proper regular model $\mathcal{C}/O_{K^{nr}}$ of $C_{K^{nr}}$ as previously presented in this section. In this subsection we show how the Galois group $\textrm{Gal}(K^{nr}/K)$ acts on the $O_{K^{nr}}$ -scheme $\mathcal{C}$ . Moreover, we describe the induced action of $G_k$ on the special fibre $\mathcal{C}_s/k^{\textrm{s}}$ , and give defining equations for the principal components of $\mathcal{C}_s$ compatibly with the action.

For any $l=1,\dots,m$ , recall the notation $f_l(x)=f(x+w_l)\in K^{nr}[x]$ and $C^{w_l}/K^{nr}\;:\;y^2=f_l(x)$ . Fix $\sigma \in G_K$ . Let $l,h=1,\dots,m$ such that $\sigma ({\mathfrak{s}}_l)={\mathfrak{s}}_h$ . Then $\sigma (f_l)=f_h$ . Now, let $\mathfrak{t}\in \Sigma _{C_{K^{nr}}}^{w_l}$ be a proper cluster. Then $\sigma (\mathfrak{t})\in \Sigma _{C_{K^{nr}}}^{w_h}$ and $\rho _{\mathfrak{t}}=\rho _{\sigma (\mathfrak{t})}$ . It follows that most of the quantities attached to $\mathfrak{t}$ , such as those of Definition 4.6, are the same for $\sigma (\mathfrak{t})$ , for example $\epsilon _{\mathfrak{t}}=\epsilon _{\sigma (\mathfrak{t})}$ . In particular, if $M$ is a matrix associated to $\mathfrak{t}$ then $M$ is associated to $\sigma (\mathfrak{t})$ as well. So $\sigma (\mathcal{F}_M^l)=\mathcal{F}_{M}^h$ . Finally, as $\sigma (\prod _{o\neq l}(x+w_{lo})^{-1})=\prod _{o\neq h}(x+w_{ho})^{-1}$ we also have $\sigma (T_M^l)=T_{M}^h$ .

Hence the natural $K^{nr}$ -isomorphism $C^{w_h}\xrightarrow{\sigma }C^{w_l}$ induces $O_{K^{nr}}$ -isomorphisms of schemes

(10) \begin{equation} \mathcal{C}_\Delta ^{w_h}\xrightarrow{\sigma }\mathcal{C}_\Delta ^{w_l},\qquad\mathring{\mathcal{C}}_\Delta ^{w_h}\xrightarrow{\sigma }\mathring{\mathcal{C}}_\Delta ^{w_l},\qquad U_M^h\xrightarrow{\sigma }U_M^l. \end{equation}

Via the glueing morphisms (4), these maps describe the action of $G_K$ on $\mathcal{C}$ .

We want to study the action of $G_k$ on the special fibre of $\mathcal{C}$ more in detail. Let $\sigma \in \textrm{Gal}(K^{nr}/K)$ and let $\bar \sigma \in G_k$ corresponding to $\sigma$ via the canonical isomorphism $\textrm{Gal}(K^{nr}/K)\simeq G_k$ . Let $l,h$ and $\mathfrak{t}$ as above. In Section 5.6 we described closed $1$ -dimensional subschemes composing $\mathring{\mathcal{C}}_{\Delta,s}^{w_l}$ and the morphisms induced by the glueing maps. Recall the polynomials introduced in Definition 5.12. From (10) we get

\begin{align*} \bar \sigma ({\overline{g_{{\mathfrak{s}}_l,w_l}^0}})={\overline{g_{{\mathfrak{s}}_h, w_h}^0}},\quad \bar \sigma ({\overline{g_{\mathfrak{t},w_l}}})={\overline{g_{\sigma (\mathfrak{t}),w_h}}},\quad \bar \sigma ({\overline{g_l|_{L_{\mathfrak{t}}^{w_l}}}})={\overline{g_h|_{L_{\sigma (\mathfrak{t})}^{w_h}}}}. \end{align*}

From the equality (7) we obtain $\bar \sigma (f_{\mathfrak{t},w_l})=f_{\sigma (\mathfrak{t}),w_h}$ . Note that the previous relations can also be recovered directly from the definitions.

Lemma 5.23. Let $w_{\mathfrak{t}}$ be the rational centre of $\mathfrak{t}$ fixed above. Then

1. (i) $\overline{g_{\mathfrak{t},w_{\mathfrak{t}}}},\overline{f_{\mathfrak{t},w_{\mathfrak{t}}}}\in k_{\mathfrak{t}}[X]$ ;

2. (ii) $\overline{g_{\mathfrak{t},w_{\mathfrak{t}}}}=\overline{g_{\mathfrak{t},w_l}}$ and $ \overline{f_{\mathfrak{t},w_{\mathfrak{t}}}}(X)=\overline{f_{\mathfrak{t},w_l}}(X+\overline{u_{w_{\mathfrak{t}} w_l}})$ where $\overline{u_{w_{\mathfrak{t}} w_l}}=\tfrac{w_{\mathfrak{t}}-w_l}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ ;

Proof. For any rational centre $w$ of $\mathfrak{t}$ , let $u_{\mathfrak{t},w}=c_f\textstyle \prod _{r\in \mathfrak{R}\setminus \mathfrak{t}} (w-r)$ as in Definition 5.12. Note that $u_{\mathfrak{t},w}/\pi ^{v(u_{\mathfrak{t},w})}$ is independent of $w$ since

\begin{align*} v((w_{\mathfrak{t}}-r)-(w-r))=v(w_{\mathfrak{t}}-w)\geq \rho _{\mathfrak{t}}\gt v(w_{\mathfrak{t}}-r) \end{align*}

for any $r\in \mathfrak{R}\setminus \mathfrak{t}$ . Then $\overline{g_{\mathfrak{t},w_{\mathfrak{t}}}}=\overline{g_{\mathfrak{t},w_l}}$ . If $\bar \sigma \in \textrm{Gal}(k^{\textrm{s}}/k_{\mathfrak{t}})$ , i.e. $\sigma \in \textrm{Gal}(K^{nr}/K_{\mathfrak{t}})$ , then

\begin{align*} \bar \sigma (\overline{g_{\mathfrak{t},w_{\mathfrak{t}}}})=\bar \sigma (\overline{g_{\mathfrak{t},w_l}})=\overline{g_{\mathfrak{t},w_h}}=\overline{g_{\mathfrak{t},w_{\mathfrak{t}}}}. \end{align*}

In particular $\overline{g_{\mathfrak{t},w_{\mathfrak{t}}}}\in k_{\mathfrak{t}}[X]$ .

Since $u_{\mathfrak{t},w}/\pi ^{v(u_{\mathfrak{t},w})}$ is independent of $w$ we also have

\begin{align*} \overline{f_{\mathfrak{t},w_{\mathfrak{t}}}}(X^{b_{\mathfrak{t}}})= \overline{f_{\mathfrak{t},w_l}}((X+{\overline{u_{w_{\mathfrak{t}} w_l}}})^{b_{\mathfrak{t}}}). \end{align*}

Suppose $\rho _{\mathfrak{t}}\in \mathbb{Z}$ . Then $b_{\mathfrak{t}}=1$ and so the equality above implies $\overline{f_{\mathfrak{t},w_{\mathfrak{t}}}}(X)=\overline{f_{\mathfrak{t},w_l}}(X+{\overline{u_{w_{\mathfrak{t}} w_l}}})$ . Suppose $\rho \notin \mathbb{Z}$ . Then $v(w-w_{\mathfrak{t}})\gt \rho _{\mathfrak{t}}$ for any rational centre $w$ of $\mathfrak{t}$ as $v(w-w_{\mathfrak{t}})\in \mathbb{Z}$ and $v(w-w_{\mathfrak{t}})\geq \rho _{\mathfrak{t}}$ . Hence ${\overline{u_{w_{\mathfrak{t}} w_l}}}=0$ . Thus $\overline{f_{\mathfrak{t},w_{\mathfrak{t}}}}(X^{b_{\mathfrak{t}}})=\overline{f_{\mathfrak{t},w_l}}(X^{b_{\mathfrak{t}}})$ , which implies $\overline{f_{\mathfrak{t},w_{\mathfrak{t}}}}(X)=\overline{f_{\mathfrak{t},w_l}}(X)=\overline{f_{\mathfrak{t},w_l}}(X+{\overline{u_{w_{\mathfrak{t}} w_l}}})$ . If $\bar \sigma \in \textrm{Gal}(k^{\textrm{s}}/k_{\mathfrak{t}})$ , i.e. $\sigma \in \textrm{Gal}(K^{nr}/K_{\mathfrak{t}})$ , then

\begin{align*} \bar \sigma (\overline{f_{\mathfrak{t},w_{\mathfrak{t}}}})(X)=\bar \sigma (\overline{f_{\mathfrak{t},w_l}})(X+\bar \sigma ({\overline{u_{w_{\mathfrak{t}} w_l}}}))=\overline{f_{\mathfrak{t},w_h}}(X+{\overline{u_{w_{\mathfrak{t}} w_h}}})=\overline{f_{\mathfrak{t},w_{\mathfrak{t}}}}(X), \end{align*}

and so $\overline{f_{\mathfrak{t},w_{\mathfrak{t}}}}\in k_{\mathfrak{t}}[X]$ .

Remark 5.24. Note that the polynomials $\overline{f_{\mathfrak{t},w_{\mathfrak{t}}}}$ , $\overline{g_{\mathfrak{t},w_{\mathfrak{t}}}}$ and $\overline{g_{{\mathfrak{s}}_h,w_h}^0}$ equal the polynomials $\overline{f_{\mathfrak{t}}}$ , $\overline{g_{\mathfrak{t}}}$ and $\overline{g_{{\mathfrak{s}}_h}^0}$ given in Definition 4.22 .

Let $V=V_{\mathfrak{t}}^{w_l}$ and consider the subscheme $X_V\times{\mathbb{P}}_V$ of $\mathcal{C}_s$ given by $V$ , where ${\mathbb{P}}_V$ is a chain of ${\mathbb{P}}^1$ s and $X_V\;:\;\{\overline{g_{\mathfrak{t},w_l}}=0\}$ over ${\mathbb{G}}_{m,k^{\textrm{s}}}$ . If ${\mathfrak{s}}_h\subset \mathfrak{t}$ , then the glueing map $U_M^h\rightarrow U_M^l$ induces the identity $\phi _V^{hl}\;:\;X_{V_{\mathfrak{t}}^{w_h}}\xrightarrow{=}X_{V_{\mathfrak{t}}^{w_l}}$ . Define $X_{\mathfrak{t}}\subseteq{\mathbb{G}}_{m,k^{\textrm{s}}}$ given by $g_{\mathfrak{t},w_{\mathfrak{t}}}=0$ . By Lemma 5.23, $\phi _V^o\;:\;X_{\mathfrak{t}}\xrightarrow{\simeq } X_{V_{\mathfrak{t}}^{w_o}}$ , for $o=h,l$ , and this isomorphism is compatible with the Galois action and the glueing maps, that is $\sigma \circ \phi _V^h=\phi _V^l\circ \sigma$ and $\phi _V^{hl}\circ \phi _V^h=\phi _V^l$ as morphisms on $X_{\mathfrak{t}}$ .

When $V_0=V_0^{w_l}$ we can consider the subscheme $X_{V_0}\times{\mathbb{P}}_{V_0}$ given by $V_0$ , where ${\mathbb{P}}_{V_0}$ is a chain of ${\mathbb{P}}^1$ s and $X_{V_0}\;:\;\{\overline{g_{{\mathfrak{s}}_l,w_l}}=0\}$ over ${\mathbb{G}}_{m,k^{\textrm{s}}}$ . Since $X_{V_0}\times{\mathbb{P}}_{V_0}$ is never glued to any other component there is no need to choose a different model for it.

Let $L=L_{\mathfrak{t}}^{w_l}$ and consider the subscheme $X_{L}^W\times{\mathbb{P}}_{L}$ given by $L$ , where ${\mathbb{P}}_{L}$ is a chain of ${\mathbb{P}}^1$ s and $X_{L}^W\;:\;\{\overline{f_{\mathfrak{t},w_l}}=0\}$ over ${\mathbb{A}}^1_{k^{\textrm{s}}}$ . If ${\mathfrak{s}}_h\subset \mathfrak{t}$ , then the isomorphism $\phi _L^{hl}\;:\;X_{L_{\mathfrak{t}}^{w_h}}^W\xrightarrow{\simeq } X_{L_{\mathfrak{t}}^{w_l}}^W$ given by the glueing map $U_M^h\rightarrow U_M^l$ is induced by the ring isomorphism $k^{\textrm{s}}[X]\rightarrow k^{\textrm{s}}[X]$ , sending $X\mapsto X+\overline{u_{w_hw_l}}$ , where $\overline{u_{w_hw_l}}=\frac{w_h-w_l}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ . Define $X_{\mathfrak{t}}^W\subseteq{\mathbb{A}}^1_{k^{\textrm{s}}}$ given by ${\overline{f_{\mathfrak{t},w_{\mathfrak{t}}}}}=0$ . By Lemma 5.23, the map $X\mapsto X+\overline{u_{w_{\mathfrak{t}} w_l}}$ induces an isomorphism $\phi _L^o\;:\;X_{\mathfrak{t}}^W\xrightarrow{\simeq } X_{L_{\mathfrak{t}}^{w_o}}^W$ , for $o=h,l$ , compatible with the Galois action and the glueing morphisms, that is $\sigma \circ \phi _L^h=\phi _L^l\circ \sigma$ and $\phi _L^{hl}\circ \phi _L^h=\phi _L^l$ as morphisms on $X_{\mathfrak{t}}^W$ .

Recall the definitions of $\hat{\mathfrak{t}}^W$ and ${\mathbb{G}}_{\mathfrak{t},w_l}\subseteq{\mathbb{A}}_{k^{\textrm{s}}}^1$ given in Definition 5.15 and the definition of $\mathfrak{t}$ given in Definition 4.22. Note that by Lemma 5.22,

\begin{align*} \hat{\mathfrak{t}}^W=\{\mathfrak{s}\in \Sigma _{K^{nr}}\cup \{\varnothing \}\mid \mathfrak{s}\lt \mathfrak{t}\}\setminus \{\{r\}\in \Sigma _{K^{nr}}\mid r\notin W\}. \end{align*}

Fix $c=0,\dots,b_{\mathfrak{t}}-1$ such that $1/b_{\mathfrak{t}}-c\rho _{\mathfrak{t}}\in \mathbb{Z}$ . For any rational centre $w\in K^{nr}$ of $\mathfrak{t}$ define $\hat{f}_{\mathfrak{t},w}\in k^{\textrm{s}}[X,Y]$ by

\begin{align*} \hat{f}_{\mathfrak{t},w}(X)=\prod _{\mathfrak{s}\in \hat{\mathfrak{t}}^W}(X-{\overline{u_{w_{\mathfrak{s}} w}}})^{\tfrac{|\mathfrak{s}|}{b_{\mathfrak{t}}}-c\epsilon _{\mathfrak{t}}}{\overline{f_{\mathfrak{t},w}}}(X), \end{align*}

where ${\overline{u_{w_{\mathfrak{s}} w}}}=\tfrac{w_{\mathfrak{s}}-w}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ ( $w_{\mathfrak{s}}=w_l$ if $\mathfrak{s}=\varnothing$ ). Let $L=L_{\mathfrak{t}}^{w_l}$ , $F=F_{\mathfrak{t}}^{w_l}$ and $M=M_{L,0}$ . It follows from Lemma 5.17 that the scheme $\Gamma _{\mathfrak{t}}^{w_l}={X}_F\cap U_M^{l}$ is given by $Y^{D_{\mathfrak{t}}}=\hat{f}_{\mathfrak{t},w_l}(X)$ as a subscheme of ${\mathbb{G}}_{\mathfrak{t},w_l}\times{\mathbb{A}}_{k^{\textrm{s}}}^1$ . We then obtain $\bar \sigma (\hat{f}_{\mathfrak{t},w_l})=\hat{f}_{\sigma (\mathfrak{t}),w_h}$ from the action (10) of $\sigma$ on $U_M^l$ .

Lemma 5.25. With the notation above,

1. (i) $\hat{f}_{\mathfrak{t},w_{\mathfrak{t}}}\in k_{\mathfrak{t}}[X]$ ;

2. (ii) $\hat{f}_{\mathfrak{t},w_{\mathfrak{t}}}(X)=\hat{f}_{\mathfrak{t},w_l}(X+{\overline{u_{w_{\mathfrak{t}} w_l}}})$ where $\overline{u_{w_{\mathfrak{t}} w_l}}=\tfrac{w_{\mathfrak{t}}-w_l}{\pi ^{\rho _{\mathfrak{t}}}}\mod \pi$ ;

Proof. If $\mathfrak{s}\in \mathfrak{t}$ , then $\sigma (\mathfrak{s})\in{(\sigma (\mathfrak{t}))}$ and $\bar \sigma ({\overline{u_{w_{\mathfrak{s}} w}}})={\overline{u_{w_{\sigma (\mathfrak{s})}\sigma (w)}}}$ for any rational centre $w$ of $\mathfrak{t}$ . Hence $\hat{f}_{\mathfrak{t},w_{\mathfrak{t}}}\in k_{\mathfrak{t}}[X]$ and $\bar \sigma (\hat{f}_{\mathfrak{t},w_l})=\hat{f}_{\sigma (\mathfrak{t}),w_h}$ by Lemma 5.23(i),(iii). Since ${\overline{u_{w_{\mathfrak{s}} w_{\mathfrak{t}}}}}={\overline{u_{w_{\mathfrak{s}} w_l}}}-{\overline{u_{w_{\mathfrak{t}} w_l}}}$ , Lemma 5.23(ii) implies $\hat{f}_{\mathfrak{t},w_{\mathfrak{t}}}(X)=\hat{f}_{\mathfrak{t},w_l}(X+{\overline{u_{w_{\mathfrak{t}} w_l}}})$ .

Define $\Gamma _{\mathfrak{t}}^{w_{\mathfrak{t}}}\subset{\mathbb{G}}_{\mathfrak{t},w_{\mathfrak{t}}}\times{\mathbb{A}}^1_{k^{\textrm{s}}}$ given by $Y^{D_{\mathfrak{t}}}=\hat{f}_{\mathfrak{t},w_{\mathfrak{t}}}$ . Suppose ${\mathfrak{s}}_h\subset \mathfrak{t}$ , and let $\phi _{\mathfrak{t}}^{hl}\;:\;\Gamma _{\mathfrak{t}}^{w_h}\simeq \Gamma _{\mathfrak{t}}^{w_l}$ be the isomorphism coming from the glueing map $U_M^h\rightarrow U_M^l$ induced by the ring homomorphism $X\mapsto X+\overline{u_{w_hw_l}}$ . By Lemma 5.25, the map $X\mapsto X+\overline{u_{w_{\mathfrak{t}} w_l}}$ induces an isomorphism $\phi _{\mathfrak{t}}^o:\Gamma _{\mathfrak{t}}^{w_{\mathfrak{t}}}\simeq \Gamma _{\mathfrak{t}}^{w_o}$ , for $o=h,l$ , which is compatible with the Galois action and the glueing maps, that is $\sigma \circ \phi _{\mathfrak{t}}^h=\phi _{\mathfrak{t}}^l\circ \sigma$ and $\phi _{\mathfrak{t}}^{hl}\circ \phi _{\mathfrak{t}}^h=\phi _{\mathfrak{t}}^l$ as morphisms on $\Gamma _{\mathfrak{t}}^{w_{\mathfrak{t}}}$ . Therefore $\Gamma _{\mathfrak{t}}$ is isomorphic to the smooth completion of $\Gamma _{\mathfrak{t}}^{w_{\mathfrak{t}}}$ , and so it is given by $Y^{D_{\mathfrak{t}}}=\tilde{f}_{\mathfrak{t}}(X)$ , where $\tilde{f}_{\mathfrak{t}}(X)=\prod _{\mathfrak{s}\in \mathfrak{t}}(X-{\overline{u_{w_{\mathfrak{s}} w_{\mathfrak{t}}}}}){\overline{f_{\mathfrak{t},w_{\mathfrak{t}}}}}(X)$ is the polynomial given in Definition 4.22.

6. Integral differentials

Let $C$ be a hyperelliptic curve of genus $g\geq 1$ defined over $K$ by a Weierstrass equation $y^2=f(x)$ . It is well-known that the $K$ -vector space of global sections of the sheaf of differentials of $C$ , namely $H^0(C,\Omega ^1_{C/K})$ , is spanned by the basis

\begin{align*} \underline{\omega }=\left \{\frac{dx}{2y},x\frac{dx}{2y},\dots,x^{g-1}\frac{dx}{2y}\right \}. \end{align*}

Let $\mathcal{C}$ be a regular model of $C$ over $O_K$ and consider its canonical (or dualising) sheaf $\omega _{\mathcal{C}/O_K}$ . The free $O_K$ -module of its global sections $H^0(\mathcal{C},\omega _{\mathcal{C}/O_K})$ can be viewed as an $O_K$ -lattice in $H^0(C,\Omega ^1_{C/K})$ (see [9, Corollary 9.2.25(a)]). The aim of this section is to present a basis of $H^0(\mathcal{C},\omega _{\mathcal{C}/O_K})$ as an $O_K$ -linear combination of the elements in $\underline{\omega }$ under the assumptions of Theorem 4.23. Note that by [9, Corollary 9.2.25(b)] the problem is independent of the choice of model $\mathcal{C}$ but it does depend on the choice of the equation $y^2=f(x)$ since the basis $\underline{\omega }$ does. Throughout this section let $C$ and $\mathcal{C}/O_K$ be as above.

If $C$ is $\Delta _v$ -regular, [1, Theorem 8.12] gives an $O_K$ -basis of $H^0(\mathcal{C},\omega _{\mathcal{C}/O_K})$ , as required. We rephrase it in terms of rational cluster invariants, by using results of Section 3 and Lemma 4.12.

Theorem 6.1. Suppose $C$ has an almost rational cluster picture and is $y$ -regular, and all proper clusters $\mathfrak{s}\in \Sigma _C$ have same rational centre $w\in K$ . Let ${\mathfrak{s}}_1\subset \dots \subset{\mathfrak{s}}_n=\mathfrak{R}$ be the proper clusters in $\Sigma _C^{\textrm{rat}}$ . For every $j=0,\dots,g-1$ , define

\begin{align*} i_j\;:\!=\;\min \{i\in \{1,\dots,n\}\mid 2(j+1)\lt |{\mathfrak{s}}_i|\} \end{align*}

and

\begin{align*} e_j\;:\!=\;\tfrac{1}{2}\epsilon _{{\mathfrak{s}}_{i_j}}-(j+1)\rho _{{\mathfrak{s}}_{i_j}}. \end{align*}

Then the differentials

\begin{align*} \mu _j=\pi ^{\lfloor e_j \rfloor } (x-w)^j\frac{dx}{2y}\qquad j=0,\dots,g-1, \end{align*}

form an $O_K$ -basis of $H^0(\mathcal{C},\omega _{\mathcal{C}/O_K})$ .

Proof. Let $C^w:y^2=f(x+w)$ be the hyperelliptic curve isomorphic to $C$ through the change of variable $y\mapsto y,$ $x\mapsto x+w$ . By Corollary 3.25 and Lemma 4.12, the curve $C^w$ is $\Delta _v$ -regular. Since $\Sigma _C^{\textrm{rat}}$ consists of the proper clusters in $\Sigma _C^w$ , Lemma 4.3 and [1, Theorem 8.12] implies that

\begin{align*} \mu _j=\pi ^{\lfloor e_j \rfloor } x^j\frac{dx}{2y}\qquad j=0,\dots,g-1, \end{align*}

form an $O_K$ -basis of $H^0(\mathcal{C},\omega _{\mathcal{C}/O_K})$ as a lattice in $H^0(C^w,\Omega ^1_{C^w/K})$ (that is if $\mathcal{C}$ is regarded as a model of $C^w$ ). Changing variables concludes the proof.

Suppose now $C$ has an almost rational cluster picture and is $y$ -regular. Let $\Sigma _C^{\textrm{min}}$ be the set of rationally minimal clusters and let $W=\{w_{\mathfrak{s}}\,\mid \,\mathfrak{s}\in \Sigma _C^{\textrm{min}}\}$ be a corresponding set of rational centres, such that all clusters in $\Sigma _C^{w_{\mathfrak{s}}}$ are proper. For every proper cluster $\mathfrak{t}\in \Sigma _C^{\textrm{rat}}$ , choose a minimal cluster $\mathfrak{s}\subseteq \mathfrak{t}$ and set $w_{\mathfrak{t}}\;:\!=\;w_{\mathfrak{s}}$ . Consider the regular model $\mathcal{C}/O_K$ of $C$ of Theorem 4.18, constructed in Section 5 by glueing the open subschemes $\mathring{\mathcal{C}}_\Delta ^{w}$ of $\mathcal{C}_\Delta ^w$ for $w\in W$ . We want to describe the canonical morphism $C\rightarrow \mathcal{C}$ . Write $W=\{w_1,\dots,w_m\}$ as in Section 5. For any $h=1,\dots,m$ , let $\mathfrak{t}\in \Sigma _C^{w_h}$ be a proper cluster and let $M$ be a matrix associated to $\mathfrak{t}$ . Let $C^{w_h}:y^2=f(x+w_h)$ and

\begin{align*} y^2-f(x+w_h)\stackrel{M}{=}Y^{n_Y}Z^{n_Z}\mathcal{F}_M^h(X,Y,Z). \end{align*}

Then, on the affine chart $X_M$ the projection $C\rightarrow \mathcal{C}_\Delta ^{w_h}$ is induced by

\begin{align*} \frac{R}{\left (\mathcal{F}_M^h(X,Y,Z)\right )}\xrightarrow{M}\frac{K[(x')^{\pm 1},(y')^{\pm 1}]}{\left ((y')^2-f(x'+w_h)\right )}\xrightarrow{\simeq }\frac{K[x^{\pm 1},y^{\pm 1}]}{\left ( y^2-f(x)\right )}, \end{align*}

where $(X,Y,Z)=(x',y',\pi )\bullet M$ and $(x',y')=(x-w_h,y)$ . In particular it follows that $(X,Y,Z)=(x-w_h,y,z)\bullet M$ and so

\begin{align*} \begin{pmatrix} x-w_h \\ y \\ \pi \end{pmatrix}= \begin{pmatrix} X^{\tilde{m}_{11}}Y^{\tilde{m}_{21}}Z^{\tilde{m}_{31}}\\ X^{\tilde{m}_{12}}Y^{\tilde{m}_{22}}Z^{\tilde{m}_{32}}\\ X^{\tilde{m}_{13}}Y^{\tilde{m}_{23}}Z^{\tilde{m}_{33}} \end{pmatrix} =\begin{pmatrix} X \\ Y \\ Z \end{pmatrix}\bullet M^{-1}. \end{align*}

For a proper cluster $\mathfrak{t}\in \Sigma _C^{\textrm{rat}}$ recall the definitions of $\Gamma _{\mathfrak{t}}$ and $m_{\mathfrak{t}}$ .

Proposition 6.2. Let $\mathfrak{t}\in \Sigma _C^{\textrm{rat}}$ be a proper cluster. Then 5

\begin{align*} &\textrm{ord}_{\Gamma _{\mathfrak{t}}}(x-w_{\mathfrak{s}})=m_{\mathfrak{t}}\rho _{\mathfrak{t}},\\ &\textrm{ord}_{\Gamma _{\mathfrak{t}}}\tfrac{dx}{2y}=-m_{\mathfrak{t}}\left (\tfrac{1}{2}\epsilon _{\mathfrak{t}}-\rho _{\mathfrak{t}}-1\right )-1. \end{align*}

for every proper cluster $\mathfrak{s}\in \Sigma _C^{\textrm{rat}}$ , $\mathfrak{s}\subseteq \mathfrak{t}$ .

Proof. Let $g(x,y)=y^2-f(x)$ . Let $W=\{w_1,\dots,w_m\}$ as above. Let $h\in \{1,\dots,m\}$ such that $w_h=w_{\mathfrak{s}}$ . Let $F=F_{\mathfrak{t}}^{w_h}$ , $V=V_{\mathfrak{t}}^{w_h}$ , $M=M_{V,0}$ and let $X,Y,Z$ be the transformed variables $(X,Y,Z)=(x-w_{\mathfrak{s}},y,\pi )\bullet M$ . Define $\mathcal{H}(X,Y,Z)=\pi - X^{\tilde{m}_{13}}Y^{\tilde{m}_{23}}Z^{\tilde{m}_{33}}$ , and $\mathcal{G}(X,Y,Z)=g((X,Y,Z)\bullet M^{-1})$ . We have

\begin{align*} \mathcal{F}_M^h(X,Y,Z)=Y^{-n_Y}Z^{-n_Z}\mathcal{G}(X,Y,Z), \end{align*}

where $n_Z=m_{\mathfrak{t}}\epsilon _{\mathfrak{t}}$ , since $\textrm{ord}_Z(y^2)=m_{\mathfrak{t}}\epsilon _{\mathfrak{t}}$ for Lemma 5.2. Write $\mathcal{F}=\mathcal{F}_M^h$ .

Note that $d(x-w_{\mathfrak{s}})=dx$ and $(g_{w_{\mathfrak{s}}})_x^{\prime}(x-w_{\mathfrak{s}})=g_x^{\prime}(x)$ , where $g_{w_{\mathfrak{s}}}(x,y)=g(x+w_{\mathfrak{s}},y)$ . Then, by [1, 8.7],

\begin{align*} \begin{cases} (x-w_{\mathfrak{s}})g_x^{\prime}=m_{11}X\mathcal{G}_X^{\prime}+m_{12}Y\mathcal{G}_Y^{\prime}+m_{13}Z\mathcal{G}_Z^{\prime}\\ yg_y^{\prime}=m_{21}X\mathcal{G}_X^{\prime}+m_{22}Y\mathcal{G}_Y^{\prime}+m_{23}Z\mathcal{G}_Z^{\prime} \end{cases} \end{align*}

from which it follows that

\begin{align*} m_{11}yg_y^{\prime}-m_{21}(x-w_{\mathfrak{s}})g_x^{\prime}&=(m_{11}m_{22}-m_{21}m_{12})Y\mathcal{G}_Y^{\prime}-(m_{21}m_{13}-m_{11}m_{23})Z\mathcal{G}_Z^{\prime}\\ &=\tilde{m}_{33}Y\mathcal{G}_Y^{\prime}-\tilde{m}_{23}Z\mathcal{G}_Z^{\prime}. \end{align*}

We will show later that this quantity is non-zero. Moreover,

\begin{align*} \tilde{m}_{33}Y\mathcal{G}_Y^{\prime}-\tilde{m}_{23}Z\mathcal{G}_Z^{\prime}=Y^{n_Y}Z^{n_Z}\left (\tilde{m}_{33}Y\mathcal{F}_Y^{\prime}-\tilde{m}_{23}Z\mathcal{F}_Z^{\prime}+(n_Y+n_Z)\mathcal{F}\right ). \end{align*}

Recall that $X=(x-w_{\mathfrak{s}})^{m_{11}}y^{m_{21}}\pi ^{m_{31}}$ . Then $\frac{dX}{X}=m_{11}\tfrac{dx}{x-w_{\mathfrak{s}}}+m_{21}\tfrac{dy}{y}$ . Furthermore, as $0=dg=g_x^{\prime}dx+g_y^{\prime}dy$ in $\Omega _{C/K}$ , we have

\begin{align*} \frac{dX}{X}=\left (\frac{m_{11}}{x-w_{\mathfrak{s}}}-\frac{m_{21}}{y}\frac{g_x^{\prime}}{g_y^{\prime}}\right ) dx=\frac{dx}{(x-w_{\mathfrak{s}})yg_y^{\prime}}\left ( m_{11}yg_y^{\prime}-m_{21}(x-w_{\mathfrak{s}}) g_x^{\prime}\right ). \end{align*}

Therefore

(11) \begin{equation} \frac{dx}{2(x-w_{\mathfrak{s}})y^2}= \frac{dX}{XY^{n_Y}Z^{n_Z}\left (\tilde{m}_{33}Y\mathcal{F}_Y^{\prime}-\tilde{m}_{23}Z\mathcal{F}_Z^{\prime}+(n_Y+n_Z)\mathcal{F}\right )}. \end{equation}

Let $S=\textrm{Spec}\, O_K$ . Considering $X^{-1}$ as an independent variable, the scheme

\begin{align*} U=\textrm{Spec}\,\frac{O_K[Y,Z,X^{-1},X]}{(\mathcal{F},\mathcal{H},X\cdot X^{-1}-1)} \end{align*}

defines a complete intersection in ${\mathbb{A}}_S^4$ . Furthermore, $U$ is an open subscheme of $\mathcal{C}_\Delta ^{w_h}\cap X_M$ that restricted to ${\mathbb{A}}_S^4\setminus \{T_M^h(X,Y,Z)=0\}$ equals $U_M^h$ . In particular, $U$ is integral. From Section 5.5 it follows that $U_{\mathfrak{t}}=U\cap \{Z=0\}$ is a dense open subset of $ X_{F}$ . Recall that $ X_{F}$ is an open subscheme of $\Gamma _{\mathfrak{t}}$ . Hence it suffices to prove the proposition for $U_{\mathfrak{t}}$ instead of $\Gamma _{\mathfrak{t}}$ ([9, Lemma 9.2.17(a)]). Since $X$ and $Y$ are units and $Z$ vanishes to order $1$ on $U_{\mathfrak{t}}$ , Lemma 5.2 implies that

(12) \begin{equation} \textrm{ord}_{U_{\mathfrak{t}}}(x-w_{\mathfrak{s}})=\tilde{m}_{31}=m_{\mathfrak{t}}\rho _{\mathfrak{t}},\,\,\,\, \textrm{ord}_{U_{\mathfrak{t}}}y=\tilde{m}_{32}=m_{\mathfrak{t}}\tfrac{\epsilon _{\mathfrak{t}}}{2},\,\,\,\, \textrm{ord}_{U_{\mathfrak{t}}}\pi =\tilde{m}_{33}=m_{\mathfrak{t}}. \end{equation}

Recall that $U$ is integral and that $U_\eta$ is isomorphic to an open subscheme of $C$ . Then $U_\eta$ is smooth. Hence, by [9, Corollary 6.4.14(b)], the sheaf $\omega _{\mathcal{C}/O_K}$ is generated on $U$ by $\mathcal{E}^{-1}dX$ where

(13) \begin{equation} \mathcal{E}\;:\!=\;\left |\begin{matrix}\mathcal{F}^{\prime}_Y&\mathcal{F}^{\prime}_Z&\mathcal{F}_{X^{-1}}^{\prime}\\ H_Y^{\prime}&H_Z^{\prime}&\mathcal{F}_{X^{-1}}^{\prime}\\ 0&0&X\end{matrix}\right |=-\pi X Y^{-1}Z^{-1}\left ( \tilde{m}_{33}Y\mathcal{F}_Y^{\prime}-\tilde{m}_{23}Z\mathcal{F}_Z^{\prime}\right ), \end{equation}

if $\mathcal{E}$ is non-zero. Suppose it is; we are going to prove it later. Thus, as $\mathcal{F}=0$ on $U$ , we have

\begin{align*} \textrm{ord}_{U_{\mathfrak{t}}}\frac{dx}{2(x-w_{\mathfrak{s}})y^2}&=\textrm{ord}_{U_{\mathfrak{t}}}\tfrac{dX}{XY^{n_Y}Z^{n_Z}\left ( \tilde{m}_{33}Y\mathcal{F}_Y^{\prime}-\tilde{m}_{23}Z\mathcal{F}_Z^{\prime}\right )}&&\text{from (11)}\\ &=\textrm{ord}_{U_{\mathfrak{t}}}\left (\pi Y^{-n_Y-1}Z^{-n_Z-1}\mathcal{E}^{-1}dX\right )&&\text{from (13)}\\ &=m_{\mathfrak{t}}-n_Z-1=m_{\mathfrak{t}}\left ( -\epsilon _{\mathfrak{t}}+1\right )-1&&\text{from (12).} \end{align*}

Then it follows from (12) that

\begin{align*} \textrm{ord}_{U_{\mathfrak{t}}}\frac{dx}{2y}=m_{\mathfrak{t}}\left (\rho _{\mathfrak{t}}+\tfrac{1}{2}\epsilon _{\mathfrak{t}}\right )+\textrm{ord}_{U_{\mathfrak{t}}}\frac{dx}{2(x-w_{\mathfrak{s}})y^2}=m_{\mathfrak{t}}\left (-\tfrac{1}{2}\epsilon _{\mathfrak{t}}+\rho _{\mathfrak{t}}+1\right )-1. \end{align*}

It remains to show that $\mathcal{E}$ does not equal $0$ on $U$ . Suppose it does. Then from the computations above, it follows that $m_{11}y g_y^{\prime}-m_{21}(x-w_{\mathfrak{s}})g_x^{\prime}=0$ in $K(C)$ . Since $m_{21}$ equals either $1$ or $2$ by Lemma 5.2, it follows that there exists a non-zero $c\in K$ , such that

\begin{align*} m_{11} y g_y^{\prime}-m_{21}(x-w_{\mathfrak{s}})g_x^{\prime}+c g=0 \end{align*}

( $c\in K$ from degree analysis). Then $cf(x)=m_{21}(x-w_{\mathfrak{s}})f'(x)$ . Note that $m_{21}$ is non-zero as $\textrm{char}(K)\neq 2$ . But then a contradiction follows since $f$ is a separable polynomial of degree $\geq 3$ .

In the following two theorems we describe a basis of integral differentials of $C$ . We use Definitions/Notations 3.1, 3.3, 3.2, 3.8, 3.9, 3.26, 4.6, 4.10 in the statements.

Theorem 6.3. Let $C/K$ be a hyperelliptic curve of genus $g\geq 1$ defined by the Weierstrass equation $y^2=f(x)$ and let $\mathcal{C}/O_K$ be a regular model of $C$ . Suppose $C$ has an almost rational cluster picture and is $y$ -regular. For $i=0,\dots,g-1$ inductively

1. (i) define $e_i\;:\!=\;\displaystyle \max _{\mathfrak{t}\in \Sigma _C^{\textrm{rat}}}\bigg \{\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}-\sum _{j=0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}\bigg \}$ ;

2. (ii) choose a maximal element ${\mathfrak{s}}_i$ of $\displaystyle \bigg \{\mathfrak{t}\in \Sigma _C^{\textrm{rat}}\mid \,e_i=\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}-\sum _{j=0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}\bigg \}$ freely.

Then the differentials

\begin{align*} \mu _i=\pi ^{\lfloor e_i\rfloor }\prod _{j=0}^{i-1}(x-w_{{\mathfrak{s}}_j})\frac{dx}{2y},\qquad i=0,\dots,g-1, \end{align*}

form an $O_K$ -basis of $H^0(\mathcal{C},\omega _{\mathcal{C}/O_K})$ .

Proof. Since $H^0(\mathcal{C},\omega _{\mathcal{C}/O_K})$ is independent of the choice of regular model, we consider $\mathcal{C}$ to be the model described in Theorem 4.18 and constructed in Section 5.

We first show that the differentials $\mu _i$ are global sections of $\omega _{\mathcal{C}/O_K}$ . It suffices to prove they are regular along all components $\Gamma _{\mathfrak{t}}$ , where $\mathfrak{t}\in \Sigma _C^{\textrm{rat}}$ proper. Indeed for the construction of $\mathcal{C}$ and the definition of the $e_i$ ’s, the differentials $\mu _i$ are regular along all other components of $\mathcal{C}_s$ by Theorem 6.1. Fix $i=1,\dots,g-1$ and let $j=0,\dots,i-1$ . Let $\mathfrak{t}\in \Sigma _C^{\textrm{rat}}$ be a proper cluster. If ${\mathfrak{s}}_j\subseteq \mathfrak{t}$ then

\begin{align*} \textrm{ord}_{\Gamma _{\mathfrak{t}}}(x-w_{{\mathfrak{s}}_j})=m_{\mathfrak{t}}\rho _{\mathfrak{t}}=m_{\mathfrak{t}}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}, \end{align*}

by Proposition 6.2. If $\mathfrak{t}\subsetneq{\mathfrak{s}}_j$ then $w_{\mathfrak{t}}$ is a rational centre of ${\mathfrak{s}}_j$ . Hence

\begin{align*} v(w_{\mathfrak{t}}-w_{{\mathfrak{s}}_j})\geq \min _{r\in \mathfrak{t}}\min \{v( r-w_{\mathfrak{t}}),v( r-w_{{\mathfrak{s}}_j})\}\geq \min \{\rho _{\mathfrak{t}},\rho _{{\mathfrak{s}}_j}\}=\rho _{{\mathfrak{s}}_j}=\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}. \end{align*}

Therefore Proposition 6.2 implies

\begin{align*} \textrm{ord}_{\Gamma _{\mathfrak{t}}}(x-w_{{\mathfrak{s}}_j})&\geq \min \{\textrm{ord}_{\Gamma _{\mathfrak{t}}}(x-w_{\mathfrak{t}}),\textrm{ord}_{\Gamma _{\mathfrak{t}}}(w_{\mathfrak{t}}-w_{{\mathfrak{s}}_j})\}\\ &\geq \min \{m_{\mathfrak{t}}\rho _{\mathfrak{t}},m_{\mathfrak{t}}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}\}=m_{\mathfrak{t}}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}. \end{align*}

If ${\mathfrak{s}}_j\nsubseteq \mathfrak{t}$ and $\mathfrak{t}\nsubseteq{\mathfrak{s}}_j$ then from Lemma 3.18 it follows that

\begin{align*} \textrm{ord}_{\Gamma _{\mathfrak{t}}}(x-w_{{\mathfrak{s}}_j})=\min \{m_{\mathfrak{t}}\rho _{\mathfrak{t}},m_{\mathfrak{t}}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}\}=m_{\mathfrak{t}}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}. \end{align*}

as $\rho _{\mathfrak{t}}\gt \rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}$ . Thus we have proved that

(14) \begin{equation} \textrm{ord}_{\Gamma _{\mathfrak{t}}}(x-w_{{\mathfrak{s}}_j})\geq m_{\mathfrak{t}}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}},\qquad \text{where the equality holds if $\mathfrak{t}\not \subset{\mathfrak{s}}_j$.} \end{equation}

Hence it follows from Proposition 6.2 that

\begin{align*} \textrm{ord}_{\Gamma _{\mathfrak{t}}}\mu _i\geq m_{\mathfrak{t}}\bigg (\lfloor e_i\rfloor +\sum _{j=0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}-\frac{\epsilon _{\mathfrak{t}}}{2}+\rho _t+1\bigg )-1. \end{align*}

But

\begin{align*} \lfloor e_i\rfloor \geq \bigg \lfloor \frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}-\sum _{j=0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}\bigg \rfloor \gt \frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}-\sum _{j=0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}-1, \end{align*}

then $\textrm{ord}_{\Gamma _{\mathfrak{t}}}\mu _i\gt -1$ , that implies $\textrm{ord}_{\Gamma _{\mathfrak{t}}}\mu _i\geq 0$ , as required.

Now we need to show that the differentials $\mu _i$ span $H^0(\mathcal{C},\omega _{\mathcal{C}/O_K})$ , that is the lattice they span is saturated in the global sections of $\omega _{\mathcal{C}/O_K}$ . Suppose not. Then there exist $I\subseteq \{0,\dots,g-1\}$ and $u_i\in O_K^\times$ for $i\in I$ such that the differential $\frac{1}{\pi }\sum _{i\in I}u_i\mu _i$ is regular along $\Gamma _{\mathfrak{t}}$ , for every proper cluster $\mathfrak{t}\in \Sigma _C^{\textrm{rat}}$ . First we want to show that for any $i_1,i_2=0,\dots,g-1$ with $i_1\lt i_2$ , one has ${\mathfrak{s}}_{i_2}\not \subset{\mathfrak{s}}_{i_1}$ . Suppose by contradiction that ${\mathfrak{s}}_{i_2}\subsetneq{\mathfrak{s}}_{i_1}$ . Then

\begin{align*} e_{i_2}&\geq \frac{\epsilon _{{\mathfrak{s}}_{i_1}}}{2}-\rho _{{\mathfrak{s}}_{i_1}}-\sum _{j=0}^{i_2-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_1}}= e_{i_1}-\rho _{{\mathfrak{s}}_{i_1}}-\sum _{j=i_1+1}^{i_2-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_1}}\geq e_{i_1}-\rho _{{\mathfrak{s}}_{i_1}}-\sum _{j=i_1+1}^{i_2-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_2}}\\ &\geq \frac{\epsilon _{{\mathfrak{s}}_{i_2}}}{2}-\rho _{{\mathfrak{s}}_{i_2}}-\sum _{j=0}^{i_1-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_2}}-\rho _{{\mathfrak{s}}_{i_1}}-\sum _{j=i_1+1}^{i_2-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_2}}=\frac{\epsilon _{{\mathfrak{s}}_{i_2}}}{2}-\sum _{j=0}^{i_2}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_2}}=e_{i_2}. \end{align*}

Therefore

\begin{align*} \max _{\mathfrak{t}\in \Sigma _C^{\textrm{rat}}}\bigg \{\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}-\sum _{j=0}^{i_2-1}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}\bigg \}=e_{i_2}=\frac{\epsilon _{{\mathfrak{s}}_{i_1}}}{2}-\rho _{{\mathfrak{s}}_{i_1}}-\sum _{j=0}^{i_2-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_1}}, \end{align*}

and this means that ${\mathfrak{s}}_{i_1}$ is a possible choice for the $i_2$ th cluster ${\mathfrak{s}}_{i_2}$ . But ${\mathfrak{s}}_{i_2}\subsetneq{\mathfrak{s}}_{i_1}$ , so the $i_2$ th cluster should have been ${\mathfrak{s}}_{i_1}$ , a contradiction.

Let $I_0\subseteq I$ be the set of indices $i$ such that $\gamma _i\;:\!=\;e_i-\lfloor e_i\rfloor$ is maximal. Let $i_0=\min I_0$ and let $\Gamma _0=\Gamma _{{\mathfrak{s}}_{i_0}}$ . Since ${\mathfrak{s}}_{i_0}\not \subset{\mathfrak{s}}_j$ , for all $j=0,\dots,i_0-1$ , from (14) it follows that

\begin{align*} m\;:\!=\;\textrm{ord}_{\Gamma _0}\frac{1}{\pi }\mu _{i_0}&=-m_{{\mathfrak{s}}_{i_0}}\gamma _{i_0}+m_{{\mathfrak{s}}_{i_0}}\bigg (e_{i_0}-\frac{\epsilon _{{\mathfrak{s}}_{i_0}}}{2}+\rho _{{\mathfrak{s}}_{i_0}}+\sum _{j=0}^{i_0-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_0}}\bigg )-1\\ &=-m_{{\mathfrak{s}}_{i_0}}\gamma _{i_0}-1\lt 0. \end{align*}

Furthermore,

\begin{align*} \textrm{ord}_{\Gamma _0}\frac{1}{\pi }\mu _{i}&\geq -m_{{\mathfrak{s}}_{i_0}}\gamma _{i}+m_{{\mathfrak{s}}_{i_0}}\bigg (e_{i}-\frac{\epsilon _{{\mathfrak{s}}_{i_0}}}{2}+\rho _{{\mathfrak{s}}_{i_0}}+\sum _{j=0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_0}}\bigg )-1\\ &\geq -m_{{\mathfrak{s}}_{i_0}}\gamma _{i}-1\geq -m_{{\mathfrak{s}}_{i_0}}\gamma _{i_0}-1=m, \end{align*}

for all $i\in I$ . Let $J\;:\!=\;\{i\in I\mid \textrm{ord}_{\Gamma _0}\frac{1}{\pi }\mu _i=m\}$ . Then $J\neq \varnothing$ since $i_0\in J$ . The order of the differential $\frac{1}{\pi }\sum _{i\in J} u_i\mu _i$ along $\Gamma _0$ must be $\gt m$ . Let $i\in I$ . From the computations above $i\in J$ if and only if

1. (i) $\textrm{ord}_{\Gamma _0}(x-w_{{\mathfrak{s}}_j})=m_{{\mathfrak{s}}_{i_0}}\rho _{{\mathfrak{s}}_{i_0}\wedge{\mathfrak{s}}_j}$ for all $j=0,\dots,i-1$ . Equivalently, if ${\mathfrak{s}}_j\supsetneq{\mathfrak{s}}_{i_0}$ for some $j\lt i$ , then $v(w_{{\mathfrak{s}}_{i_0}}-w_{{\mathfrak{s}}_j})=\rho _{{\mathfrak{s}}_{i_0}\wedge{\mathfrak{s}}_j}$ .

2. (ii) $e_i=\frac{\epsilon _{{\mathfrak{s}}_{i_0}}}{2}-\rho _{{\mathfrak{s}}_{i_0}}-\sum _{j=0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_0}}$ . In particular, if ${\mathfrak{s}}_i\subseteq{\mathfrak{s}}_{i_0}$ , then ${\mathfrak{s}}_i={\mathfrak{s}}_{i_0}$ .

3. (iii) $\gamma _i=\gamma _{i_0}$ . Equivalently, $i\in I_0$ .

Therefore $J\subseteq I_0$ , $i_0=\min J$ and

\begin{align*} \lfloor e_i\rfloor -\lfloor e_{i_0}\rfloor =e_i-\gamma _i-e_{i_0}+\gamma _{i_0}=e_i-e_{i_0}=-\sum _{j=i_0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_0}}, \end{align*}

for all $i\in J$ . Hence

\begin{align*} \frac{1}{\pi }\sum _{i\in{J}}u_i\mu _i=\frac{1}{\pi }\mu _{i_0}\bigg (\sum _{i\in{J}}\frac{u_i}{\pi ^{\sum _{j=i_0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_0}}}}\prod _{j=i_0}^{i-1}(x-w_{{\mathfrak{s}}_j})\bigg ), \end{align*}

and since $\textrm{ord}_{\Gamma _0}\frac{1}{\pi }\mu _{i_0}=m\lt 0$ we must have

(15) \begin{equation} \textrm{ord}_{\Gamma _0}\bigg (\sum _{i\in{J}}\frac{u_i}{\pi ^{\sum _{j=i_0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_0}}}}\prod _{j=i_0}^{i-1}(x-w_{{\mathfrak{s}}_j})\bigg )\gt 0. \end{equation}

For any $j\lt i\in J$ , with $i_0\leq j$ we have ${\mathfrak{s}}_j\not \subset{\mathfrak{s}}_{i_0}$ . Therefore either ${\mathfrak{s}}_j={\mathfrak{s}}_{i_0}$ or ${\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_0}\supsetneq{\mathfrak{s}}_{i_0}$ . In the latter case,

\begin{align*} \textrm{ord}_{\Gamma _0}(x-w_{{\mathfrak{s}}_{i_0}})=m_{{\mathfrak{s}}_{i_0}}\rho _{{\mathfrak{s}}_{i_0}}\gt m_{{\mathfrak{s}}_{i_0}}\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_0}}=\textrm{ord}_{\Gamma _0}(x-w_{{\mathfrak{s}}_j}). \end{align*}

It follows from (15) that

\begin{align*} \textrm{ord}_{\Gamma _0}\bigg (\sum _{i\in{J}}v_i\frac{(x-w_{{\mathfrak{s}}_{i_0}})^{\beta _i}}{{\pi ^{\beta _i\rho _{{\mathfrak{s}}_{i_0}}}}}\bigg )\gt 0, \end{align*}

where $J_i=\{j\in I\mid i_0\leq j\lt i \mbox{ and }{\mathfrak{s}}_j\neq{\mathfrak{s}}_{i_0}\}$ , $v_i=u_i\prod _{j\in J_i}\frac{w_{{\mathfrak{s}}_{i_0}}-w_{{\mathfrak{s}}_{j}}}{\pi ^{\rho _{{\mathfrak{s}}_j\wedge{\mathfrak{s}}_{i_0}}}}\in O_K^\times$ , and $\beta _i=|\{i_0,\dots,i-1\}\setminus J_i|$ .

To find a contradiction, we will use the explicit description of a dense open affine subset of $\Gamma _0$ . Let $W=\{w_1,\dots,w_m\}$ be the set of rational centres of the rationally minimal clusters for $C$ fixed at the beginning of the section. Let $w_h\in W$ such that $w_h=w_{{\mathfrak{s}}_{i_0}}$ , and let $L=L_{{\mathfrak{s}}_{i_0}}^{w_h}$ , $M=M_{L,0}$ , and consider

\begin{align*} U_M^h\cap \{Z=0\}=\textrm{Spec}\,\frac{R[T_M^h(X,Y,Z)^{-1}]}{\left (\mathcal{F}_M^h(X,Y,Z),Z\right )}\subset \Gamma _{\mathfrak{t}}, \end{align*}

dense open subscheme of $\Gamma _{\mathfrak{t}}$ . From Lemma 5.2,

\begin{align*} \sum _{i\in{J}}v_i\frac{(x-w_h)^{\beta _i}}{{\pi ^{\beta _i\rho _{{\mathfrak{s}}_{i_0}}}}} =\sum _{i\in{J}}v_i X^{\beta _i/b_{{\mathfrak{s}}_{i_0}}}, \end{align*}

which is a unit since the polynomial $\mathcal{F}_M^h(X,Y,Z)$ in $\{Z=0\}$ is of the form $Y^2-G(X)$ or $Y-G(X)$ for some non-constant $G(X)\in K[X]$ (for more details see Lemma 5.17). This gives a contradiction and concludes the proof.

Assume now $C_{K^{nr}}$ has an almost rational cluster picture and is $y$ -regular as in Theorem 4.23. Since $|\Sigma _C|$ is finite, there exists a finite unramified extension $F/K$ such that $C_F$ has an almost rational cluster picture and is $y$ -regular. Denote by $O_F$ the ring of integers of $F$ . Let $\Sigma _F=\Sigma _{C_F}^{\textrm{rat}}$ . Fix a rational centre $w_{\mathfrak{s}}\in F$ for every rationally minimal cluster $\mathfrak{s}\in \Sigma _F$ . For all non-minimal clusters $\mathfrak{t}\in \Sigma _F$ choose a rational centre $w_{\mathfrak{t}}=w_{\mathfrak{s}}$ for some rationally minimal cluster $\mathfrak{s}\subseteq \mathfrak{t}$ . In this setting the next theorem gives a basis of integral differentials of $C$ .

Theorem 6.4. Let $C/K$ be a hyperelliptic curve of genus $g\geq 1$ defined by the Weierstrass equation $y^2=f(x)$ and let $\mathcal{C}/O_K$ be a regular model of $C$ . Suppose there exists a finite unramified extension $F/K$ such that $C_F$ has an almost rational cluster picture and is $y$ -regular. For $i=0,\dots,g-1$ inductively

1. (i) define $e_i\;:\!=\;\displaystyle \max _{\mathfrak{t}\in \Sigma _F}\bigg \{\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}-\sum _{j=0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}\bigg \}$ ;

2. (ii) choose a maximal element ${\mathfrak{s}}_i$ of $\displaystyle \bigg \{\mathfrak{t}\in \Sigma _F\mid \,e_i=\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}-\sum _{j=0}^{i-1}\rho _{{\mathfrak{s}}_j\wedge \mathfrak{t}}\bigg \}$ freely.

Then the differentials

\begin{align*} \mu _i=\pi ^{\lfloor e_i\rfloor }\cdot \textrm{Tr}_{F/K}\bigg (\beta \prod _{j=0}^{i-1}(x-w_{{\mathfrak{s}}_j})\bigg )\frac{dx}{2y},\qquad i=0,\dots,g-1, \end{align*}

form an $O_K$ -basis of $H^0(\mathcal{C},\omega _{\mathcal{C}/O_K})$ .

Proof. First note that without loss of generality we can suppose $F/K$ Galois. Moreover, since $F/K$ is unramified, $\textrm{Gal}(F/K)\simeq \textrm{Gal}(\mathfrak{f}/k)$ , where $\mathfrak{f}$ is the residue field of $F$ , and so the existence of $\beta$ is guaranteed by the surjectivity of $\textrm{Tr}_{\mathfrak{f}/k}$ . Let $\mathcal{C}$ be the minimal regular model of $C$ over $O_K$ . By [9, Proposition 10.1.17], the base extended scheme $\mathcal{C}_{O_F}$ is the minimal regular model of $C_F$ over $O_F$ . Let $\mu _0^F,\dots,\mu _{g-1}^F$ be the basis of integral differentials of $C_F$ given by Theorem 6.3.

Suppose $\mu _0^{\prime},\dots,\mu _{g-1}^{\prime}$ is a basis of integral differentials of $C_F$ that, for any $\sigma \in \textrm{Gal}(F/K)$ and any $j=0,\dots,g-1$ , satisfies

(16) \begin{equation} \sigma (\mu _j^{\prime})=\mu _j^{\prime}+\sum _{0\leq i\lt j}\lambda _{ij}\mu _i^{\prime}, \end{equation}

for some $\lambda _{ij}\in O_F$ (depending on $\sigma$ ). Note that $\mu _0^F,\dots,\mu _{g-1}^F$ is in fact such a basis. We want to prove that, for any $j=0,\dots,g-1$ , the differentials

(17) \begin{equation} \mu _0^{\prime},\dots,\mu _{j-1}^{\prime},\textrm{Tr}_{F/K}(\beta \mu _j^{\prime}),\mu _{j+1}^{\prime},\dots, \mu _{g-1}^{\prime} \end{equation}

still form a basis of $H^0(\mathcal{C}_F,\omega _{\mathcal{C}_F/O_F})$ satisfying condition (16). From equation (16) it follows that

\begin{align*} \textrm{Tr}_{F/K}(\beta \mu _j^{\prime})=\sum _{\sigma \in \textrm{Gal}(F/K)}\sigma (\beta )\sigma (\mu _j^{\prime})=\textrm{Tr}_{F/K}(\beta )\mu _j^{\prime}+\sum _{i\lt j}\lambda _{ij}^{\prime}\mu _i^{\prime}, \end{align*}

for some $\lambda _{ij}^{\prime}\in O_F$ . Since $\textrm{Tr}_{F/K}(\beta )\in O_K^\times$ , the differentials in (17) form a basis of $H^0(\mathcal{C}_F,\omega _{\mathcal{C}_F/O_F})$ satisfying condition (16).

Since $\mu _0^F,\dots,\mu _{g-1}^F$ satisfies (16), by induction it follows that

\begin{align*} \textrm{Tr}_{F/K}(\beta \mu _0^F),\dots,\textrm{Tr}_{F/K}(\beta \mu _{g-1}^F) \end{align*}

is a basis of $H^0(\mathcal{C}_F,\omega _{\mathcal{C}_F/O_F})$ . Proposition B.2 concludes the proof.

We conclude this section with an application of Theorem 6.3.

Example 6.5. Let $p$ be a prime number and let $a\in \mathbb{Z}_p$ , $b\in \mathbb{Z}_p^\times$ such that the polynomial $x^2+ax+b$ is not a square modulo $p$ . Let $C$ be the hyperelliptic curve over $\mathbb{Q}_p$ of genus $4$ described by the equation $y^2=f(x)$ , where $f(x)=(x^6+ap^4x^3+bp^8)((x-p)^3-p^{11})$ . We have already shown in Examples 3.32 and 4.25 that $C$ satisfies the hypothesis of Theorem 6.3 and has rational cluster picture

We choose rational centres for the minimal clusters $\mathfrak{t}_3$ and $\mathfrak{t}_4$ : $w_{\mathfrak{t}_3}=0$ and $w_{\mathfrak{t}_4}=p$ . Since $\mathfrak{R}=\mathfrak{t}_3\wedge \mathfrak{t}_4$ , we can set either $w_{\mathfrak{R}}=w_{\mathfrak{t}_3}$ or $w_{\mathfrak{R}}=w_{\mathfrak{t}_4}$ . Let us fix $w_{\mathfrak{R}}=w_{\mathfrak{t}_3}=0$ . Then to choose ${\mathfrak{s}}_0,{\mathfrak{s}}_1,{\mathfrak{s}}_2,{\mathfrak{s}}_3$ as in Theorem 6.3 we draw the following table:

The numbers in red indicate that ${\mathfrak{s}}_0=\mathfrak{t}_4$ , ${\mathfrak{s}}_1={\mathfrak{s}}_2=\mathfrak{t}_3$ and ${\mathfrak{s}}_3=\mathfrak{R}$ . Thus the differentials

\begin{align*} \mu _0=p^4\cdot \frac{dx}{2y},\quad \mu _1=p^3\cdot (x-p)\frac{dx}{2y},\quad \mu _2=p\cdot (x-p)x\frac{dx}{2y},\quad \mu _3=(x-p)x^2\frac{dx}{2y} \end{align*}

form a $\mathbb{Z}_p$ -basis of $H^0(\mathcal{C},\omega _{\mathcal{C}/\mathbb{Z}_p})$ , for any regular model $\mathcal{C}/\mathbb{Z}_p$ of $C$ .