ON $\mu _{n}$-ACTIONS ON K3 SURFACES IN POSITIVE CHARACTERISTIC

Abstract

In characteristic $0$ , symplectic automorphisms of K3 surfaces (i.e., automorphisms preserving the global $2$ -form) and non-symplectic ones behave differently. In this paper, we consider the actions of the group schemes $\mu _{n}$ on K3 surfaces (possibly with rational double point [RDP] singularities) in characteristic p, where n may be divisible by p. We introduce the notion of symplecticness of such actions, and we show that symplectic $\mu _{n}$ -actions have similar properties, such as possible orders, fixed loci, and quotients, to symplectic automorphisms of order n in characteristic $0$ . We also study local $\mu _n$ -actions on RDPs.

1 Introduction

K3 surfaces are proper smooth surfaces X with $\Omega ^2_X \cong \mathcal {O}_X$ and $H^1(X, \mathcal {O}_X) = 0$ . The first condition implies that X admits an everywhere nonvanishing $2$ -form, and such a $2$ -form is unique up to scalar. An automorphism of a K3 surface is called symplectic if it preserves the global $2$ -form. It is known that symplectic and non-symplectic automorphisms behave very differently.

For example, Nikulin [Ni, §§4 and 5] proved that quotients of K3 surfaces in characteristic $0$ by a symplectic action of a finite group G has only rational double points (RDPs) as singularities and that the minimal resolutions of the quotients are again K3 surfaces. Moreover, he determined the number of fixed points (which are always isolated) if G is cyclic. To the contrary, the quotients by non-symplectic actions of finite groups are never birational to K3 surfaces; instead, they are birational to either Enriques surfaces or rational surfaces.

These results hold in characteristic $p> 0$ provided p does not divide the order of G (see Theorem 5.1), but are no longer true for order p automorphisms in characteristic p. In this case, the notion of symplecticness is useless, since any such automorphism is automatically symplectic (since there are no nontrivial pth root of unity in characteristic p) and, for small p, there exist examples of automorphisms with non-K3 quotients (see [DK1], [DK2]).

In this paper, we consider actions of the finite group schemes $\mu _{n}$ (n may be divisible by p) on RDP K3 surfaces, by which we mean surfaces with RDP singularities whose minimal resolutions are K3 surfaces. It is essential to allow RDPs since smooth K3 surfaces never admit actions of $\mu _p$ (see Remark 2.2). We introduce the notion of symplecticness and fixed points of such actions (Definitions 2.5 and 2.7). Then we prove the following properties, which are parallel to the properties of automorphisms of order not divisible by the characteristic.

Theorem 1.1 (Theorems 6.1 and 6.2)

Let X be an RDP K3 surface in characteristic p, equipped with a $\mu _{n}$ -action. If the action is symplectic, then the quotient $X/\mu _{n}$ is an RDP K3 surface. If $n = p$ and the action is non-symplectic, then the quotient $X/\mu _{p}$ is an RDP Enriques surface if the action is fixed-point-free (which is possible only if $p = 2$ ), and otherwise it is a rational surface.

Theorem 1.2 (Theorems 7.1 and 8.2)

• • There exists an RDP K3 surface X in characteristic p equipped with a $\mu _{p}$ -action if and only if $p \leq 19$ .

• • If X is an RDP K3 surface X in characteristic p equipped with a $\mu _{n}$ -action, then $\phi (n) \leq 20$ , in particular $n \leq 66$ . Moreover, for each p, we determine the set of n for which such an action exists.

• • For each p, there exists an RDP K3 surface X in characteristic p equipped with a symplectic $\mu _{n}$ -action if and only if $n \leq 8$ , and we determine the number of fixed points.

To prove the main results, we first study (in §§3 and 4) $\mu _n$ -actions on local rings of surfaces at smooth points and RDPs. We define the notion of symplecticness of such actions (Definitions 3.1 and 4.1) and prove the following result.

Theorem 1.3 (Theorem 4.6 and Propositions 4.7 and 4.9)

Let X be the localization at a closed point w of an RDP surface in characteristic p equipped with a $\mu _{p}$ -action. Let ${\pi } : {X}\to {X/\mu _p}$ be the quotient morphism.

• • If w is not fixed by the action, then $\pi (w)$ is either a smooth point or an RDP.

• • If w is fixed and the action is symplectic at w, then w is an isolated fixed point and $\pi (w)$ is an RDP.

• • If w is an isolated fixed point and the action is non-symplectic at w, then $\pi (w)$ is a non-RDP singularity.

We classify the possible actions in the non-fixed case (Table 3) and the symplectic case (Table 4).

Moreover, we also give a partial classification of local $\mu _{p^e}$ - and $\mu _n$ -actions (Propositions 4.12 and 4.13) and a complete classification of local symplectic $\mu _n$ -actions (Proposition 4.14). We hope that these local results would have applications other than K3 surfaces.

The results on $\mu _n$ -quotients, orders of symplectic $\mu _n$ -actions, and orders of $\mu _n$ -actions on K3 surfaces are discussed in §§6–8, respectively.

In §9, we give several examples of $\mu _{n}$ -actions on K3 surfaces.

Throughout the paper, we work over an algebraically closed field k of $\operatorname {\mathrm {char}} k = p \geq 0$ . Varieties are separated integral k-schemes of finite type (not necessarily proper or smooth), and surfaces are two-dimensional varieties. We denote the smooth locus of a variety X by $X^{\mathrm {sm}}$ .

2 Preliminaries

2.1 K3 surfaces and rational double points

RDP singularities of surfaces are precisely the canonical surface singularities that are not smooth. They are classified into types $A_n$ ( $n \geq 1$ ), $D_n$ ( $n \geq 4$ ), and $E_n$ ( $n = 6,7,8$ ) by their dual graph of the exceptional curves of the minimal resolution, which are Dynkin diagrams of type $A_n$ , $D_n$ , or $E_n$ . The number n is equal to the number of the exceptional curves, and we say that the RDP is of index n. The dual graph determines the formal isomorphism class of an RDP except in certain cases in characteristics $2,3,5$ . For the exceptional cases, we use Artin’s notation $D_n^r$ and $E_n^r$ (see [A2]).

We recall the classification, given by Bombieri and Mumford [BM2], of proper smooth surfaces X with numerically trivial canonical divisor $K_X$ : they consist of four classes, with the characterizing properties as reported in Table 1. Here, $b_i = \dim H_{\mathrm {\acute et}}^i(X, \mathbb Q_l)$ is the ith l-adic Betti number for an auxiliary prime $l \neq \operatorname {\mathrm {char}} k$ . Enriques and (quasi-)hyperelliptic surfaces in characteristics $2$ and $3$ may have unusual values of $\dim H^1(\mathcal {O}_X)$ and $\operatorname {\mathrm {ord}}(K_X)$ .

Table 1 Surfaces with numerically trivial canonical divisors.

The distinction between hyperelliptic and quasi-hyperelliptic surfaces is not important in this paper. Furthermore, the choice of the origin of an abelian surface is not important.

Definition 2.1. RDP surfaces are surfaces that have only RDPs as singularities (if any). Hence, any smooth surface is an RDP surface by definition.

RDP K3 surfaces are proper RDP surfaces whose minimal resolutions are (smooth) K3 surfaces. We similarly define RDP abelian, RDP Enriques, and RDP (quasi-)hyperelliptic surfaces.

However, since abelian surfaces and (quasi-)hyperelliptic surfaces do not admit smooth rational curves, any RDP abelian or RDP (quasi-)hyperelliptic surface is smooth.

Remark 2.2. Smooth K3 surfaces in characteristic $p> 0$ admit no nontrivial global vector fields ([RS, Th. 7], [Ny1, Cor. 3.5]), and hence admit no nontrivial actions of $\mu _p$ (or $\alpha _p$ ). However, RDP K3 surfaces may admit such actions.

Proposition 2.3. For any RDP surface X, the pullback by the morphism $X^{\mathrm {sm}} \cong \tilde X \setminus E \hookrightarrow \tilde X$ to the minimal resolution $\tilde X$ of X induces an isomorphism $H^0(X^{\mathrm {sm}}, (\Omega ^2_X)^{\otimes n}) \cong H^0(\tilde X, (\Omega ^2_{\tilde X})^{\otimes n})$ , where E is the exceptional divisor. Nonvanishing forms on one side correspond to nonvanishing ones on the other side.

Proof. This follows from the following local version applied repeatedly.

Proposition 2.4. Let $(A,\mathfrak {m})$ be the localization at a closed point of an RDP surface. Then $H^0(\operatorname {\mathrm {Spec}} A \setminus \{\mathfrak {m}\}, \Omega ^2_{A/k})$ is a free A-module of rank 1. If A is smooth, then this space is isomorphic to $H^0(\operatorname {\mathrm {Spec}} A, \Omega ^2_{A/k})$ . If A is an RDP and $(A',\mathfrak {m}')$ is the localization at a closed point of $\operatorname {\mathrm {Bl}}_{\mathfrak {m}} A$ , then any generator of the above space extends to a generator of $H^0(\operatorname {\mathrm {Spec}} A' \setminus \{\mathfrak {m}'\}, \Omega ^2_{A'/k})$ .

Proof. If A is smooth, then $\Omega ^2_{A/k}$ is free of rank $1$ and the assertion is clear. Suppose A is an RDP. Then it is a hypersurface isolated singularity, and it is well known that for such singularities, the canonical divisor is trivial, and then the former assertion follows. Since an RDP is a canonical singularity, the pullback of the canonical divisor to $\operatorname {\mathrm {Bl}}_{\mathfrak {m}} A$ is also trivial, and hence the latter assertion follows.

2.2 Group schemes of multiplicative type

Recall that we are working over an algebraically closed field k.

We consider finite commutative group schemes G of multiplicative type over k. This means that G is of the form $\prod _j \mu _{n_j}$ for some positive integers $n_j$ . The Cartier dual $G^{\vee } = \operatorname {\mathrm {\mathcal {H}\mathit {om}}}(G, \mathbb G_m)$ of G is a finite étale group scheme and can be identified with the finite group $G^{\vee }(k)$ of k-valued points. Using this finite commutative group $G^{\vee }$ , we have the following explicit description: $G = \operatorname {\mathrm {Spec}} k[t_{i}]_{i \in G^{\vee }} / (t_{i} t_{j} - t_{i+j}, t_0 - 1)$ , with the group operations $m \colon G \times G \to G$ , $e \colon \operatorname {\mathrm {Spec}} k \to G$ , $i \colon G \to G$ given by $m^*(t_{i}) = t_{i} \otimes t_{i}$ , $e^*(t_{i}) = 1$ , $i^*(t_{i}) = t_{-i}$ .

An action $\alpha \colon G \times \operatorname {\mathrm {Spec}} B \to \operatorname {\mathrm {Spec}} B$ corresponds, via $\alpha ^*(b) = \sum _{i \in G^{\vee }} t_{i} \otimes \operatorname {\mathrm {pr}}_{i}(b)$ , to decompositions $B = \bigoplus _{i \in G^{\vee }} B_i$ to k-vector subspaces satisfying $B_i B_j \subset B_{i+j}$ . We say an element b or a subset of $B_i$ to be homogeneous of weight i and we write $\operatorname {\mathrm {wt}}(b) = i$ .

Such a decomposition $B = \bigoplus _{i} B_i$ naturally extends to a decomposition $\Omega ^*_{B/k} = \bigoplus _i (\Omega ^*_{B/k})_i$ satisfying $d(B_i) \subset (\Omega ^1_{B/k})_i$ and $(\Omega ^*_{B/k})_i (\Omega ^*_{B/k})_j \subset (\Omega ^*_{B/k})_{i+j}$ .

If G acts on a scheme X that is not necessarily affine but admits a covering by G-stable affine open subschemes (which is the case if, e.g., X is quasi-projective or G is local), then the G-action admits a quotient ${\pi }:{X} \to {X/G}$ , and induces decompositions $\pi _* \mathcal {O}_X = \bigoplus _{i} (\pi _* \mathcal {O}_X)_i$ , $\pi _* \Omega ^*_{X/k} = \bigoplus _{i} (\pi _* \Omega ^*_{X/k})_i$ , and $H^0(X, (\Omega ^*_{X/k})^{\otimes n}) = \bigoplus _i (H^0(X, (\Omega ^*_{X/k})^{\otimes n}))_i$ , compatible with multiplications.

If $\operatorname {\mathrm {char}} k$ does not divide the order of $G^{\vee }$ , then $B_i$ are the eigenspaces for the action of $G(k)$ with eigenvalues $i \in G^{\vee }(k) = \operatorname {\mathrm {Hom}}(G(k),k^*)$ .

If $\operatorname {\mathrm {char}} k = p> 0$ and $G^{\vee }$ is cyclic of order p (hence $G \cong \mu _p = \operatorname {\mathrm {Spec}} k[t_1]/(t_1^p - 1)$ for a choice of a generator $1$ of $G^{\vee }$ ), then giving such a decomposition is also equivalent to giving a k-derivation D on B of multiplicative type (i.e., $D^p = D$ ) under the correspondence $B_i = B^{D = i} = \{b \in B \mid D(b) = ib \}$ (this correspondence depends on the choice of a generator $1$ of $G^{\vee }$ ). Moreover, D extends to a k-linear endomorphism on $\Omega ^*_{B/k}$ satisfying $D(df) = d(D(f))$ , $D^p = D$ , and the Leibniz rule $D(\omega \wedge \eta ) = \omega \wedge D(\eta ) + D(\omega ) \wedge \eta $ .

Now, we generalize the notion of symplecticness of automorphisms to actions of group schemes like $\mu _n$ .

Definition 2.5. Let G be a finite group scheme of multiplicative type. Let X be either an abelian surface or an RDP K3 surface, equipped with an action of G. We say that the action is symplectic if the one-dimensional space $H^0(X^{\mathrm {sm}}, \Omega ^2_{X/k})$ with respect to the action of G is of weight $0$ .

Remark 2.6. Under the assumptions of Definition 2.5, suppose G is reduced. Equivalently, this means that G is a constant group scheme corresponding to a finite abelian group of order prime to p. Then, by Proposition 2.3, our symplecticness is equivalent to the symplecticness of the induced G-action on the minimal resolution $\tilde {X}$ in the usual sense (i.e., preserving the global $2$ -form). This suggests that our definition of the symplecticness of $\mu _n$ -actions is a natural generalization of that of $\mathbb Z/m\mathbb Z$ -actions (order m automorphisms) for m not divisible by $\operatorname {\mathrm {char}} k$ .

On the other hand, if $G = \mathbb Z/p\mathbb Z$ (which does not belong to the class considered in Definition 2.5), then any action of G preserves the global $2$ -form, since there are no nontrivial pth roots of unity. Thus, the usual definition of symplecticness is useless in this case. We do not know whether there is a useful notion of symplecticness in a larger class of group schemes containing $\mathbb Z/p\mathbb Z$ or $\alpha _p$ .

2.3 Derivations of multiplicative type

In this section, assume that $\operatorname {\mathrm {char}} k = p> 0$ .

Recall that, given an action of a group scheme G on a scheme X, the fixed point scheme $X^G \subset X$ is characterized by the property $X^G(T) = \operatorname {\mathrm {Hom}}_G(T,X)$ for any k-scheme T equipped with the trivial G-action. If $G = \mu _p$ and D is the corresponding derivation, we write $\operatorname {\mathrm {Fix}}(D) = X^G$ and also call it the fixed locus of D.

Definition 2.7. We say that a closed point $w \in X$ is fixed by the $\mu _{n}$ -action, or by the corresponding derivation if $n = p$ , if $w \in X^{\mu _{n}}$ .

Proposition 2.8. Let k be an algebraically closed field. Let $X = \operatorname {\mathrm {Spec}} B$ be a Noetherian affine k-scheme equipped with a $\mu _{p^e}$ -action. For each closed point $w \in X$ , the assertions (1)–(4) are equivalent. If $e = 1$ and D is the corresponding derivation, then the assertions (1)–(6) are equivalent, and if moreover X is a smooth variety, then (7) is also equivalent.

1. 1. w is a $\mu _{p^e}$ -fixed point.

2. 2. The maximal ideal $\mathfrak {m}_w$ of $\mathcal {O}_{X,w}$ is generated by homogeneous elements.

3. 3. The canonical morphism $B \to B / \mathfrak {m}_w$ is $\mu _{p^e}$ -equivariant, where $B/\mathfrak {m}_w$ is equipped with the trivial action (i.e., the decomposition concentrated on $(-)_0$ ).

4. 4. $B_i \subset \mathfrak {m}_w$ for each $i \neq 0$ .

5. 5. $D(\mathfrak {m}_w) \subset \mathfrak {m}_w$ .

6. 6. $D(\mathcal {O}_{X,w}) \subset \mathfrak {m}_w$ .

7. 7. D has singularity at w in the sense of [RS, §1].

Finally, if (1) holds, then the $\mu _{p^e}$ -action extends to the blowup $\operatorname {\mathrm {Bl}}_w X$ .

Proof. Let $B = \bigoplus _{i \in \mathbb Z/p^e\mathbb Z} B_i$ be the corresponding decomposition.

(1 $\iff $ 3) By the definition of $X^{\mu _p}$ , a closed point $w \in X$ is a (k-valued) point of $X^{\mu _p}$ if and only if $B \to B/\mathfrak {m}_w$ is compatible with the projections $\operatorname {\mathrm {pr}}_i$ to the ith summand $(-)_i$ for all i, where $B/\mathfrak {m}_w$ is equipped with the trivial decomposition.

(2 $\iff $ 3) If (3) holds, then we have $\operatorname {\mathrm {pr}}_i(\mathfrak {m}_w) \subset \mathfrak {m}_w$ for all i, and then each element x of $\mathfrak {m}_w$ is the sum of homogeneous elements $\operatorname {\mathrm {pr}}_i(x) \in \mathfrak {m}_w$ . Conversely, if $\mathfrak {m}_w$ is generated by homogeneous elements, then $\operatorname {\mathrm {pr}}_i(\mathfrak {m}_w) \subset \mathfrak {m}_w$ for all i, which implies (3).

(3 $\iff $ 4) Easy.

Assume $e = 1$ .

(2 $\iff $ 5) Assume $D(\mathfrak {m}_w) \subset \mathfrak {m}_w$ . Take a system of generators $(x_j)$ of $\mathfrak {m}_w$ . For each j, let $x_j = \sum _{i \in \mathbb F_p} x_{j,i}$ be the decomposition of $x_j$ in $B = \bigoplus _{i} B_i$ . Then $D^l(x_j) = \sum _i i^l x_{j,i}$ is also in $\mathfrak {m}_w$ . Since the matrix $(i^l)_{i,l=0}^{p-1}$ is invertible, this implies that $x_{j,i} \in \mathfrak {m}_w$ . Thus $\mathfrak {m}_w$ is generated by eigenvectors. The converse is clear.

(5 $\iff $ 6) This is clear since $\mathcal {O}_{X,w} = \mathfrak {m}_w + k$ and $D \rvert _{k} = 0$ .

(5 $\iff $ 7) Take coordinates $x_1, \ldots , x_n$ at a point w and write $D = \sum _j f_j \cdot (\partial / \partial x_j)$ . Then both of the conditions are equivalent to $(f_j) \subset \mathfrak {m}_w$ .

We show the final assertion assuming (2). If the maximal ideal $\mathfrak {m}$ is generated by homogeneous elements $x_j \in B_{i_j}$ , then, for each j, we can extend the action on the affine piece $\operatorname {\mathrm {Spec}} B[x_h/x_j]_h$ of $\operatorname {\mathrm {Bl}}_w X$ by declaring $x_h/x_j$ to be homogeneous of weight $i_h - i_j$ .

The next lemma enables us to take useful coordinates at a point not fixed by D.

Lemma 2.9. If B is a Noetherian local ring, D is a derivation of multiplicative type, and the closed point is not fixed by D, then the maximal ideal $\mathfrak {m}$ of B is generated by elements $x_1, \ldots , x_{m-1}, y$ with $\operatorname {\mathrm {wt}}(x_j) = 0$ and $\operatorname {\mathrm {wt}}(1+y) = 1$ . If $\mathfrak {m}$ is generated by n elements, then we can take $m = n$ . If $\dim B \geq 2$ , then D does not extend to a derivation of the blowup $\operatorname {\mathrm {Bl}}_{\mathfrak {m}} B$ .

Proof. Recall that a subset of $\mathfrak {m}$ generates $\mathfrak {m}$ if and only if it generates $\mathfrak {m}/\mathfrak {m}^2$ .

Take elements $x^{\prime }_1, \dots , x^{\prime }_m$ generating $\mathfrak {m}$ , and let $x^{\prime }_j = \sum _{i \in \mathbb F_p} x^{\prime }_{j,i}$ be the decompositions to eigenvectors. By assumption, there exists a pair $(j,i)$ with $x^{\prime }_{j,i} \not \in \mathfrak {m}$ . We take such $j_0,i_0$ , and we may assume $i_0 \neq 0$ . We may assume $x^{\prime }_{j_0,i_0} - 1 \in \mathfrak {m}$ . Then $y = x^{\prime }_{j_0,i_0} - 1 $ satisfies $y \in \mathfrak {m}$ and $D(y) = i_0 (y + 1)$ . We have $y \not \in \mathfrak {m}^2$ , since $D(\mathfrak {m}^2) \subset \mathfrak {m}$ . By replacing y with $(y+1)^q - 1$ for an integer q with $q i_0 \equiv 1 \pmod p$ , we may assume $i_0 = 1$ . For each j, let $x_j = \sum _i (y+1)^{-i} x^{\prime }_{j,i}$ . Then we have $D(x_j) = 0$ and, since $x_j \equiv x^{\prime }_j \pmod {(y)}$ , the elements $x_j,y$ generate $\mathfrak {m}/\mathfrak {m}^2$ and hence generate $\mathfrak {m}$ . We can omit one of the $x_j$ ’s and then the remaining elements satisfies the required conditions (after renumbering).

To show the latter assertion, it suffices to show that D does not extend to $B' := B[x_j/y]_j$ . If it extends, then we have $D(x_j/y) = -x_j(y+1)/y^2 \in B'$ , hence $x_j/y^2 \in B'$ , and then on $\operatorname {\mathrm {Spec}} B'$ we have that $y = 0$ implies $x_j/y = 0$ , which is impossible since $\dim B' \geq 2$ .

Before stating the next proposition, we recall the following notion from [RS]. Assume X is a smooth irreducible variety and D is a nontrivial derivation. Then $\operatorname {\mathrm {Fix}}(D)$ consists of its divisorial part $(D)$ and non-divisorial part $\langle D \rangle $ . If we write $D = f \sum _i g_i \frac {\partial }{\partial x_i}$ for some local coordinates $x_1, \dots , x_m$ with $g_i$ having no common factor, then $(D)$ and $\langle D \rangle $ correspond to the ideals $(f)$ and $(g_i)$ , respectively. If D is of multiplicative type with $\langle D \rangle = \emptyset $ , then it follows from Proposition 2.8 that for suitable coordinates near any fixed point, we have $D = a x_m \cdot (\partial / \partial x_m)$ and that $\operatorname {\mathrm {Fix}}(D)$ is a smooth divisor (possibly empty).

Assuming that $\operatorname {\mathrm {Fix}}(D)$ is divisorial, in which case the quotient is a smooth variety by [S, Prop. 6], the highest differential forms on smooth loci of X and $X^D$ are related in the following way.

Proposition 2.10. Let X be a smooth variety of dimension m (not necessarily proper) equipped with a nontrivial derivation D of multiplicative type such that $\operatorname {\mathrm {Fix}}(D)$ is divisorial. Let $\Delta $ be the divisor $\operatorname {\mathrm {Fix}}(D)$ . Then there is a unique collection of isomorphisms

$$\begin{align*}(\pi_* (\Omega^m_{X/k}(\Delta))^{\otimes n})_0 \cong (\Omega^m_{X^D/k}(\pi_*(\Delta)))^{\otimes n} \end{align*}$$

for all integers n, compatible with multiplication, preserving the zero loci, and sending (for $n = 1$ )

$$\begin{align*}f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m) \mapsto f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m^p) \end{align*}$$

if $f_0, \dots , f_{m-1}$ are homogeneous of weight $0$ and $f_m$ is homogeneous of some weight (not necessarily $0$ ).

In particular, if the action is fixed-point-free, then we have isomorphisms

$$ \begin{align*} (\pi_* (\Omega^m_{X/k})^{\otimes n})_0 &\cong (\Omega^m_{X^D/k})^{\otimes n} \quad \text{and} \\ H^0(X, (\Omega^m_{X/k})^{\otimes n})_0 &\cong H^0(X^D, (\Omega^m_{X^D/k})^{\otimes n}) \end{align*} $$

with the same properties.

Proof. The isomorphism for $n = 0$ is clear. It suffices to construct the isomorphism for $n = 1$ that is compatible with multiplication with $n = 0$ forms and with restriction to open subschemes.

Take a closed point $w \in X$ . Let $\varepsilon = 1$ (resp. $\varepsilon = 0$ ) if $w \not \in \Delta $ (resp. $w \in \Delta $ ). By Lemma 2.9 (resp. by [RS, Th. 1]), there are coordinates $x_1, \dots , x_m$ on a neighborhood of w with $D(x_j) = 0$ for $j < m$ and $D(x_m) = a (\varepsilon + x_m)$ for some $a \in \mathbb F_p^*$ . We define

$$ \begin{align*} \phi \colon (\pi_* (\Omega^m_{X/k}(\Delta)))_0 &\to \Omega^m_{X^D/k}(\pi_*(\Delta)) \\ f \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m) &\mapsto f \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m^p) \end{align*} $$

for f of weight $0$ (note that $dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log (\varepsilon + x_m)$ is a local generator of the left-hand side). We show that then $\phi $ sends

$$\begin{align*}f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m) \mapsto f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m^p) \end{align*}$$

for any $f_0, \dots , f_{m-1}$ and $f_m$ as in the statement. This implies that $\phi $ does not depend on the choice of the coordinates and hence that $\phi $ induces a well-defined morphism of sheaves. Then since $dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log (\varepsilon + x_m)$ (resp. $dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log (\varepsilon + x_m^p)$ ) is a local generator of $(\Omega ^m_{X/k}(\Delta ))_0$ (resp. $\Omega ^m_{X^D/k}(\pi _*(\Delta ))$ ), it follows that $\phi $ is an isomorphism and $\phi ^{\otimes n}$ are well-defined isomorphisms.

We may pass to the completion, so consider $f_h \in k[[x_1, \dots , x_m]]$ . By the assumption on the weight, we have $f_h \in k[[x_1, \dots , x_{m-1}, x_m^p]]$ for $h < m$ and $f_m \in (\varepsilon + x_m)^b k[[x_1, \dots , x_{m-1}, x_m^p]]$ for some $0 \leq b < p$ . Then we have $\partial f_h / \partial x_m = 0$ for $h < m$ and $\partial f_m / \partial x_m = b f_m/(\varepsilon + x_m)$ . Hence, we have

$$ \begin{align*} & f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m) \\ &= f_0 ((\varepsilon + x_m)/f_m) \det (\partial f_h / \partial x_j)_{1 \leq h,j \leq m} \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m) \\ &= b f_0 \det (\partial f_h / \partial x_j)_{1 \leq h,j \leq m-1} \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m) \\ &\stackrel{\phi}{\mapsto} b f_0 \det (\partial f_h / \partial x_j)_{1 \leq h,j \leq m-1} \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m^p). \end{align*} $$

On the other hand, in the invariant subalgebra $k[[x_1, \dots , x_{m-1}, x_m^p]]$ , we have $\partial f_m^p / \partial x_j = 0$ for $j < m$ and $\partial f_m^p / \partial x_m^p = b f_m^p/(\varepsilon + x_m^p)$ . Hence, we have

$$ \begin{align*} & f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m^p) \\ &= \dots = b f_0 \det (\partial f_h / \partial x_j)_{1 \leq h,j \leq m-1} \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m^p). \end{align*} $$

The assertion follows.

We will give another abstract proof of Proposition 2.10 in [Mat3, Prop. 2.12].

2.4 Global properties of derivations

Lemma 2.11. Let $C \subset \mathbb P^2$ be a quadratic curve (not necessarily irreducible nor reduced) in characteristic p, and let D be a p-closed derivation. Then $\operatorname {\mathrm {Fix}}(D) \neq \emptyset $ .

Proof. Suppose C is integral. Then $C \cong \mathbb P^1$ and the result is well known (indeed, $T_{\mathbb P^1} \cong \mathcal {O}_{\mathbb P^1}(2)$ ).

Suppose C is reducible. We may assume $C = (xy = 0)$ . Let $U = \operatorname {\mathrm {Spec}} k[x,y] / (xy)$ . Then $T_U = ( x \frac {d}{dx} - y \frac {d}{dy}) \cdot \mathcal {O}_U$ and hence the origin belongs to $\operatorname {\mathrm {Fix}}(D)$ .

Suppose C is non-reduced. We may assume $C = (X_3^2 = 0)$ . If $p \neq 2$ , then D induces a derivation $D_{\mathrm {red}}$ on $C_{\mathrm {red}} \cong \mathbb P^1$ , and we have $\operatorname {\mathrm {Fix}}(D) \approx \operatorname {\mathrm {Fix}}(D_{\mathrm {red}}) \neq \emptyset $ . Suppose $p = 2$ . It is easy to see that $H^0(C, T_C) \stackrel {\sim }{\to } H^0(C, T_{\mathbb P^2} \rvert _{C}) \stackrel {\sim }{\leftarrow } H^0(\mathbb P^2, T_{\mathbb P^2})$ . Hence, there exist $f_1, f_2, f_3 \in H^0(\mathbb P^2, \mathcal {O}(1)) = \bigoplus _{i=1}^3 k X_i$ such that $D\bigl (\frac {X_i}{X_j}\bigr ) = \frac {f_i}{X_j} - \frac {X_i f_j}{X_j^2}$ . If $f_3 \in k X_3$ , then D induces a derivation $D_{\mathrm {red}}$ on $C_{\mathrm {red}} \cong \mathbb P^1$ , and we conclude as above. Suppose $f_3 \not \in k X_3$ . By a coordinate change, we may assume $f_3 - X_2 \in k X_3$ . Letting $x_i = X_i/X_1$ ( $i = 2,3$ ) and restricting to $\operatorname {\mathrm {Spec}} k[x_2,x_3]/(x_3^2) = (X_1 \neq 0) \subset C$ , we have $D(x_3) - x_2 \in (x_3)$ , in particular $D(x_3) \in \mathfrak {m} := (x_2, x_3)$ . If $D(x_2) \in \mathfrak {m}$ , then the origin is a fixed point. Suppose $D(x_2) \not \in \mathfrak {m}$ and $D^2 = h D$ . Then $h = D^2(x_2) / D(x_2) \in \mathcal {O}_{\mathfrak {m}}$ , and hence $D(x_2) \equiv D^2(x_3) = h D(x_3) \equiv 0 \pmod {\mathfrak {m}}$ , contradiction.

Corollary 2.12. Suppose $\mu _p$ acts on an RDP surface X and fixes an RDP w. Then the action extends to the blowup $\operatorname {\mathrm {Bl}}_w X$ and there exists a fixed point above w.

Proof. The action extends to the blowup by Proposition 2.8. Let $D'$ be the induced derivation on $\operatorname {\mathrm {Bl}}_w X$ . Let $C \subset \operatorname {\mathrm {Bl}}_w X$ be the (possibly non-reduced) exceptional divisor, which is a quadratic curve in $\mathbb P^2$ since w is an RDP. Since $D'(\mathcal I_C) \subset \mathcal I_C$ , $D'$ induces a derivation $D^{\prime }_{C}$ (of multiplicative type) on C. By Lemma 2.11, $D^{\prime }_C$ has at least one fixed point, and that point is also a fixed point of $D'$ .

Later we will also need the following Katsura–Takeda formula on rational vector fields (i.e., derivations on the fraction field $k(X)$ ). For a rational derivation D locally of the form $f^{-1} D'$ for some regular function f and (regular) derivation $D'$ , we define the divisorial and nondivisorial parts by $(D) = (D') - \operatorname {\mathrm {div}}(f)$ and $\langle D \rangle = \langle D' \rangle $ .

Proposition 2.13 [KT, Prop. 2.1]

Let X be a smooth proper surface, and let D be a nonzero rational vector field. Then we have

$$\begin{align*}\deg c_2(X) = \deg \langle D \rangle - K_X \cdot (D) - (D)^2. \end{align*}$$

3 Tame symplectic actions on RDPs

Hereafter, all action of groups and group schemes on schemes are assumed faithful.

Throughout this section, we work under the following setting. $B = \mathcal {O}_{X,w}$ is the localization of an RDP surface X over an algebraically closed field k at a closed point w (either a smooth point or an RDP), $\mathfrak {m} \subset B$ is the maximal ideal, G is a finite group acting on X, and the action restricts to $\operatorname {\mathrm {Spec}} B$ . Assume that the order of G is not divisible by $p = \operatorname {\mathrm {char}} k$ .

Definition 3.1. We say that the G-action on B is symplectic if it acts on the one-dimensional k-vector space $H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k}) \otimes _B (B/\mathfrak {m})$ trivially.

If $G = \mathbb Z/p\mathbb Z$ , then any action is symplectic (cf. Remark 2.6), and hence the notion is useless in this case.

Remark 3.2. If B is as above and the G-action is symplectic, then the rank- $1$ free B-module $H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k})$ admits a generator $\omega $ that is G-invariant. Indeed, take a generator $\omega '$ , then $\omega := (1 / \lvert G \rvert ) \sum _{g \in G} g^* \omega '$ is clearly G-invariant and it is nonvanishing, since it is nonvanishing after $\otimes (B / \mathfrak {m})$ .

Remark 3.3. If X is an RDP K3 surface and $w \in X$ is a fixed closed point, then this is consistent with the usual notion of symplecticness, since a generator of $H^0(X^{\mathrm {sm}}, \Omega ^2) \cong H^0(\tilde {X}, \Omega ^2)$ (Proposition 2.3) restricts to a generator of this one-dimensional space. Thus, the symplecticness of an automorphism of an RDP K3 surface can be checked locally at any fixed point (if there exists any). The same for abelian surfaces.

Proposition 3.4. Let B and G be as above (in particular, the order of G is not divisible by $p = \operatorname {\mathrm {char}} k$ ). Then the invariant ring $B^G$ is again the localization at a closed point of an RDP surface.

Let $\tilde {X} \to X$ be the minimal resolution at w. Then $\tilde {X}/G \to X/G$ is crepant.

Proof. Let $\omega $ be a generator of the rank- $1$ free B-module $H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k})$ . By Remark 3.2, we may assume $\omega $ is G-invariant. The action of G on X induces an action on $\tilde {X}$ , and $\omega $ extends to a regular nonvanishing $2$ -form on $\tilde {X}$ . At each closed point $w' \in \tilde {X}$ , the stabilizer $G_{w'} \subset G$ acts on $T_{w'} \tilde {X}$ via $\operatorname {\mathrm {SL}}_2(k)$ since G preserves $\omega $ . Hence, the quotient $\tilde {X}/G$ has only RDPs as singularities. Since $\omega $ is preserved by G, it induces a regular nonvanishing $2$ -form on $(\tilde {X}/G)^{\mathrm {sm}}$ , and since RDPs are canonical singularities, it extends to a regular nonvanishing $2$ -form on $\widetilde {\tilde {X}/G}$ , the minimal resolution of $\tilde {X}/G$ above w. Thus, $B^G$ is a canonical singularity, that is, either a smooth point or an RDP.

Remark 3.5. We [Mat1, Prop. 3.8] described possible symplectic actions of finite tame groups on RDPs. For actions of cyclic groups $G = \mathbb Z/n\mathbb Z$ ( $n> 1$ ), we have a complete classification: possible n and the types of X and $X/G$ are listed in Table 2.

Table 2 Tame symplectic cyclic actions on RDPs.

Remark 3.6. Singularities of quotients by order p automorphisms in characteristic $p> 0$ tends to be worse than those in characteristic $\neq p$ . For example, the quotient of a supersingular abelian surface in characteristic $2$ by the automorphism $x \mapsto -x$ is a rational surface with an elliptic singularity [Ka, Th. C].

4 $\mu _n$ -actions on RDPs and quotients

Throughout this section, we work under the following setting. $B = \mathcal {O}_{X,w}$ is the localization of an RDP surface X over an algebraically closed field k of characteristic $p \geq 0$ at a closed point w (either a smooth point or an RDP), $\mathfrak {m} \subset B$ is the maximal ideal, n is a positive integer possibly divisible by p, $\mu _n$ acts on X, and the action restricts to $\operatorname {\mathrm {Spec}} B$ . (Note that w is not necessarily fixed by $\mu _n$ .)

If $n = p> 0$ , then the corresponding derivation of multiplicative type is denoted by D.

4.1 Symplecticness of $\mu _n$ -actions

Assume w is fixed by the $\mu _n$ -action. Then the action on B induces an action on $V := H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k}) \otimes _B (B/\mathfrak {m})$ , that is, a decomposition $V = \bigoplus _{i \in \mathbb Z/n\mathbb Z} V_i$ of k-vector spaces. Since $\dim _k V = 1$ , V is equal to one of the summands. In other words, V is of some weight $i_0 \in \mathbb Z/n\mathbb Z$ .

Definition 4.1. We say that the $\mu _{n}$ -action, or the corresponding derivation if $n = p$ , on B is symplectic if V is of weight $0$ .

We say that a $\mu _{n}$ -action, or a derivation D of multiplicative type, on an RDP surface X is symplectic at a fixed closed point w if the induced action or derivation on $\mathcal {O}_{X,w}$ is symplectic in the above sense.

Remark 4.2. If $p \nmid n$ , then $\mu _n$ is (noncanonically) isomorphic to $\mathbb Z/n\mathbb Z$ and this definition is consistent with Definition 3.1.

Remark 4.3 (cf. Remark 3.2)

If B is as above and V is of weight $i_0$ , then the rank- $1$ free B-module $H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k})$ admits a generator $\omega $ of weight $i_0$ . Indeed, take a generator $\omega '$ , let $\omega ' = \sum _{i} \omega ^{\prime }_i$ be its decomposition, and write $\omega ^{\prime }_i = f_i \omega '$ with $f_i \in B$ . Since $\sum f_i = 1$ , there exists $i_1 \in \mathbb Z/n\mathbb Z$ with $f_{i_1} \in B^*$ . Then $i_0 = i_1$ and hence we can take $\omega = \omega ^{\prime }_{i_1}$ . If $n = p$ , then this means $D(\omega ) = i_0 \omega $ .

From this, it follows that if $\mu _{n}$ acts on an RDP surface, then the weight $i_0$ is a locally constant function on the fixed locus.

Remark 4.4 (cf. Remark 3.3)

If X is an RDP K3 surface and $w \in X$ is a fixed closed point, then the action is symplectic in the sense of Definition 2.5 if and only if action is symplectic at w, since a generator of $H^0(X^{\mathrm {sm}}, \Omega ^2)$ restricts to a generator of this one-dimensional space. Thus, the symplecticness of a $\mu _{n}$ -action on an RDP K3 surface can be checked locally at any fixed point (if there exists any). The same for abelian surfaces.

Lemma 4.5. Suppose the closed point w of B is fixed under the $\mu _n$ -action. Then B is generated by $2$ or $3$ homogeneous elements, respectively, if B is smooth or an RDP. Moreover:

1. 1. If B is smooth and generated by elements $x,y$ of respective weights $a,b$ , then the action is symplectic if and only if $a + b = 0$ (in $\mathbb Z/n\mathbb Z$ ).

2. 2. If B is an RDP and generated by $x,y,z$ of respective weights $a,b,c$ , then there is $d \in \mathbb Z/n\mathbb Z$ and a homogeneous power series $F \in k[[x,y,z]]$ of weight d such that $\hat {B} \cong k[[x,y,z]]/(F)$ . The action is symplectic if and only if $a + b + c = d$ .

Proof. The first assertion follows from Proposition 2.8.

(1) $H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k})$ is generated by $dx \wedge dy$ , which is of weight $a+b$ .

(2) Take an element $H \in k[[x,y,z]]$ such that $\hat {B} = k[[x,y,z]] / (H)$ , and let $H = \sum _{i \in \mathbb Z/n\mathbb Z} H_i$ be the decomposition with respect to the $\mu _n$ -action. Since $H = 0$ in $\hat {B}$ , we have $H_i = 0$ in $\hat {B}$ , and hence there are $f_i \in k[[x,y,z]]$ such that $H_i = f_i H$ . Since $\sum f_i = 1$ , there exists $d \in \mathbb Z/n\mathbb Z$ with $f_d \in k[[x,y,z]]^*$ . We can take $F = H_d$ , which is of weight d.

Then $H^0(\operatorname {\mathrm {Spec}} \hat {B} \setminus \{\mathfrak {m}\}, \Omega ^2_{\hat {B}/k})$ is generated by $\omega = F_x^{-1} dy \wedge dz = F_y^{-1} dz \wedge dx = F_z^{-1} dx \wedge dy$ (this means that the restriction of $\omega $ to the open subscheme $\operatorname {\mathrm {Spec}} \hat {B}[F_x^{-1}]$ is equal to $F_x^{-1} dy \wedge dz$ , and so on), and we have $\operatorname {\mathrm {wt}}(\omega ) = a + b + c - d$ since $\operatorname {\mathrm {wt}}(F_x^{-1}) = -(d - a)$ and $\operatorname {\mathrm {wt}}(dy \wedge dz) = b + c$ , and so on.

4.2 $\mu _p$ -actions on RDPs

As noted in §2.3, we know by [S, Prop. 6] (see also [RS, Th. 1 and Corollary]) that the quotient of a smooth variety by a $\mu _p$ -action with no isolated fixed point is smooth. We need to consider, more generally, the quotients of surfaces with RDP singularities and with isolated fixed points.

Let $\mathcal {O}_{X,w}$ and $\mu _n$ be as in the beginning of §4, and suppose $n = p$ . Let $\pi \colon X \to Y = X/\mu _p$ be the quotient morphism.

Theorem 4.6.

1. 1. Assume w is non-fixed. If w is a smooth point, then $\pi (w) \in Y$ is also a smooth point. If w is an RDP, then $\pi (w)$ is either a smooth point or an RDP. In either case, $X \times _Y \tilde {Y} \to X$ is crepant, where $\tilde {Y} \to Y$ is the minimal resolution at $\pi (w)$ .

2. 2. If w is fixed and the action is symplectic at w, then w is an isolated fixed point and $\pi (w)$ is an RDP.

3. 3. If w is an isolated fixed point and the action is non-symplectic at w, then $\pi (w)$ is a non-RDP singularity.

First, we consider non-symplectic actions on isolated fixed points.

Proof of Theorem 4.6(3)

By Proposition 2.10, we have an isomorphism

$$\begin{align*}(H^0(\operatorname{\mathrm{Spec}} \mathcal{O}_{X,w} \setminus \{w\}, \Omega^2))_0 \cong H^0(\operatorname{\mathrm{Spec}} \mathcal{O}_{Y,\pi(w)} \setminus \{\pi(w)\}, \Omega^2) \end{align*}$$

preserving the zero loci of $2$ -forms. If $\pi (w)$ is either a smooth point or an RDP, then the right-hand side has a nonvanishing $2$ -form and hence there is a nonvanishing form $\omega $ on $\operatorname {\mathrm {Spec}} \mathcal {O}_{X,w} \setminus \{w\}$ of weight $0$ . Being nonvanishing, $\omega $ is a generator of $H^0(\operatorname {\mathrm {Spec}} \mathcal {O}_{X,w} \setminus \{w\}, \Omega ^2)$ . However, this contradicts the non-symplecticness assumption.

Next, we consider non-fixed points. In fact, we can classify all possible actions and give explicit equations.

Proposition 4.7. Assume w is not fixed.

• • If w is a smooth point, then there are coordinates $x,y$ of $\mathcal {O}_{X,w}$ satisfying $D(x) = 0$ and $D(y) \neq 0$ , and hence $\mathcal {O}_{Y,\pi (w)}$ has $x,y^p$ as coordinates and in particular $\pi (w)$ is a smooth point.

• • If w is an RDP, then there is an element $F \in k[[x,y,z^p]]$ and an isomorphism $\hat {\mathcal {O}}_{X,w} \cong k[[x,y,z]] / (F)$ with $D(x) = D(y) = 0$ and $D(z) \neq 0$ , and hence $\hat {\mathcal {O}}_{Y,\pi (w)} \cong k[[x,y,z^p]] / (F)$ . Moreover, we can take F to be one in Table 3.

Table 3 Non-fixed $\mu _p$ -actions on RDPs.

Proof of Theorem 4.6(1) and Proposition 4.7

If w is a smooth point, then taking coordinates $x,y$ as in Lemma 2.9 (i.e., $D(x) = 0$ and $D(y) = 1+y$ ), we have $\hat {\mathcal {O}}_{Y,\pi (w)} \cong k[[x,y^p]]$ , and hence $\mathcal {O}_{Y,\pi (w)}$ is smooth.

Assume w is an RDP. By Lemma 2.9, we have coordinates $x,y,z$ satisfying $D(x) = D(y) = 0$ and $D(z) \neq 0$ . We have $\hat {\mathcal {O}}_{X,w} \cong k[[x,y,z]] / (F)$ for some $F \in k[[x,y,z]]$ such that $D(F) \in (F)$ , and we may assume $F \in k[[x,y,z^p]]$ . We show that, after replacing F with a multiple by a unit, and after a coordinate change of $k[[x,y,z]]$ that preserves the subring $k[[x,y,z^p]]$ , F coincides with one in Table 3. (Such coordinate changes are given by $x',y',z' \in \mathfrak {m}$ that are linearly independent in $\mathfrak {m}/\mathfrak {m}^2$ and satisfy $x',y' \in \mathfrak {m} \cap k[[x,y,z^p]]$ .) A similar classification is given in [EH+, Prop. 3.8], but they missed the case of $E_7^0$ in characteristic $3$ .

Assume the classification for the moment. Then, in each case, we observe that $\pi (w)$ is either a smooth point or an RDP, and it is straightforward to check that $X \times _Y \tilde {Y}$ is an RDP surface crepant over X. (In Table 3, the entries of the singularities of $X \times _Y \tilde {Y}$ are omitted if Y is already smooth.) For example, consider $X = \operatorname {\mathrm {Spec}} k[x,y,z]/(F)$ , $F = xy + z^{mp}$ with $m \geq 2$ . Then $X' := X \times _Y \operatorname {\mathrm {Bl}}_{\pi (w)} Y$ is covered by three affine pieces

$$ \begin{align*} X^{\prime}_1 &= \operatorname{\mathrm{Spec}} k[x, y_1, v_1, z] / (y_1 + x^{m-2} v_1^{m}, x v_1 - z^p), & y_1 &= y/x, & v_1 &= z^p/x, \\ X^{\prime}_2 &= \operatorname{\mathrm{Spec}} k[x_2, y, v_2, z] / (x_2 + y^{m-2} v_2^{m}, y v_2 - z^p), & x_2 &= x/y, & v_2 &= z^p/y, \\ X^{\prime}_3 &= \operatorname{\mathrm{Spec}} k[x_3, y_3, z] / (x_3 y_3 + z^{(m-2)p}), & x_3 &= x/z^p, & y_3 &= y/z^p. \end{align*} $$

One observes that $\operatorname {\mathrm {Sing}}(X')$ consists of two RDPs of type $A_{p-1}$ at the origins of $X^{\prime }_1$ and $X^{\prime }_2$ and, if $m \geq 3$ , one RDP of type $A_{(m-2)p-1}$ at the origin of $X^{\prime }_3$ . Repeating this, we observe that $X \times _Y \tilde {Y}$ has $m A_{p-1}$ .

Now, we show the classification. We say that F has a monomial if the coefficient of that monomial is nonzero. We also write $F = \sum _{h,i,j} a_{hij} x^h y^i z^j$ .

First, assume $p> 2$ . We may assume that the degree $2$ part $F_2$ is either $xy$ or $x^2$ . Assume $F_2 = xy$ . We may assume that F has no $xz^{j}$ and $yz^{j}$ . F must have $z^{j}$ , $j = mp$ , and then it is $A_{mp-1}$ . Then, by replacing x with $x + a_{0ij} y^{i-1} z^j$ and y with $y + a_{h0j} x^{h-1} z^j$ , and so on, we may assume that F has no $y^i z^j$ with $i> 0$ and no $x^h z^j$ with $h> 0$ . Thus, $F = u_1 xy + u_2 z^{mp}$ for some units $u_1, u_2$ , and then by replacing $x,y,F$ by suitable multiples, we obtain $F = xy + z^{mp}$ .

Assume $p> 3$ and $F_2 = x^2$ . We may assume that the degree $3$ part $F_3$ is $y^3$ . If $p \geq 7$ , it cannot be an RDP. If $p = 5$ , then F must have $z^5$ , and then it is $E_8^0$ . We have $F = u_1 x^2 + u_2 y^3 + u_3 z^5$ , and then by replacing $x,y,F$ by suitable multiples, we obtain $F = x^2 + y^3 + z^5$ . (For example, we let $F = u_3 F'$ , $x = (u_3 u_1^{-1})^{1/2} x'$ , $y = (u_3 u_2^{-1})^{1/3} y'$ . Note that we can take nth roots of units provided $p \nmid n$ .)

Assume $p = 3$ and $F_2 = x^2$ . We may assume $F_3 = y^3$ or $F_3 = z^3$ . If $F_3 = z^3$ , then F must have $y^4$ or $y^5$ , and then it is $E_6^0$ or $E_8^0$ . We may assume $a_{130} = a_{140} = 0$ by replacing x with $x + (1/2)(a_{130} y^3 + a_{140} y^4)$ , and then we transform F as above. If $F_3 = y^3$ , then F must have $yz^3$ and then it is $E_7^0$ . We eliminate $a_{1ij}$ as above, then we have $F = u_1 x^2 + u_2 y^3 + u_3 yz^3 + z^6 g(z^3)$ for some power series $g \in k[[z^3]]$ . We may assume $u_i \equiv 1 \pmod {\mathfrak {m}}$ . We eliminate g by replacing y with $y + z^3 g$ , and then we transform F as above.

Now, consider $p = 2$ . We may assume $F_2$ is one of $xy + z^2$ (if irreducible), $xy$ (if reducible but not a square), $z^2$ , or $x^2$ (square, of a linear factor containing z or not). If $F_2 = xy + z^2$ or $F_2 = xy$ , then as above, it is $A_{mp-1}$ and F becomes $xy + z^{mp}$ .

Assume $p = 2$ and $F_2 = x^2$ . If $F_3$ has $yz^2$ , then F must have $xy^m$ and then it is $D_{2m+1}^0$ . We obtain $F = u_1 x^2 + u_2 y z^2 + u_3 x y^m + z^4 g(z) + f(y) + y^{2m} g'(y)$ , where $f(y) = f_0(y)^2 + y f_1(y)^2$ is a polynomial of degree $< 2m$ , $g(z) \in k[[z]]$ , and $g'(y) \in k[[y]]$ . We may assume $u_i \equiv 1 \pmod {\mathfrak {m}}$ . We eliminate f by replacing x with $x + f_0(y)$ and z with $z + f_1(y)$ and so on. Then we eliminate g and $g'$ by replacing y and x suitably, and take multiples by units as above.

If $F_3$ has no $yz^2$ , then F must have $y^3$ and $xz^2$ and then it is $E_6^0$ . We obtain $F = u_1 x^2 + u_2 y^3 + u_3 x z^2 + a y^2 z^2 + z^4 g$ , $g = g_0(z^2) + y g_1(z^2) + y^2 g_2(z^2)$ . We may assume $u_i \equiv 1 \pmod {\mathfrak {m}}$ . We eliminate a and g by replacing y and x suitably, and then we transform F as above.

Assume $p = 2$ and $F_2 = z^2$ . Let $\overline {F_3} = (F_3 \bmod (z)) \in k[[x,y]]$ . If $\overline {F_3}$ has three distinct roots, then we may assume $\overline {F_3} = x^3 + y^3$ and then it is $D_4^0$ . We can transform F to $z^2 + x^3 + y^3$ as above, and then to $z^2 + x^2 y + x y^2$ . If $\overline {F_3}$ has two distinct roots, then we may assume $\overline {F_3} = x^2y$ and F must have $xy^m$ and then it is $D_{2m}^0$ . We obtain $F = u_1 z^2 + u_2 x^2 y + u_3 x y^m + g(x) + f(y) + y^{2m-1} g'(y)$ , where $f(y) = f_0(y)^2 + y f_1(y)^2$ is a polynomial of degree $< 2m-1$ and $g \in k[[x]]$ and $g' \in k[[y]]$ . We argue as in the case of $D_{2m+1}^0$ . If $\overline {F_3}$ has one (triple) root, then we may assume $\overline {F_3} = x^3$ and F must have $xy^3$ or $y^5$ and then it is $E_7^0$ or $E_8^0$ . We transform F as above.

Next, we consider symplectic actions on fixed points.

Lemma 4.8. Assume w is fixed, and the action is symplectic at w.

1. 1. Assume w is a smooth point. Then w is an isolated fixed point and $\pi (w)$ is an RDP of type $A_{p-1}$ . The eigenvalues of D on the cotangent space $\mathfrak {m}_w / \mathfrak {m}_w^2$ are of the form $a,-a$ for some $a \in \mathbb F_p^*$ .

2. 2. Assume w is an RDP. Let ${f}:{X' = \operatorname {\mathrm {Bl}}_w X}\to {X}$ . Then $X'$ is an RDP surface, D uniquely extends to a derivation $D'$ on $X'$ which is symplectic at every fixed point above w, and $g \colon Y' = (X')^{D'} \to Y$ is crepant.

Proof. (1) By Lemma 4.5(1), we have $D = a x \cdot (\partial / \partial x) - a y \cdot (\partial / \partial y)$ with $a \in \mathbb F_p$ for some coordinates $x,y$ , and $a \neq 0$ since D is nontrivial. Hence, w is an isolated fixed point of D. We observe that $a,-a$ are the eigenvalues of the action on the cotangent space. We have $\hat {\mathcal {O}}_{X,w}^D = k[[x^p, xy, y^p]]$ , and it is an RDP of type $A_{p-1}$ .

(2) By Remark 4.3 and assertion (1), w is an isolated fixed point. By Proposition 2.8, D uniquely extends to $D'$ on $X'$ . Let $\omega $ be a generator of $H^0(\operatorname {\mathrm {Spec}} \mathcal {O}_{X,w} \setminus \{w\}, \Omega ^2)$ with $D(\omega ) = 0$ . Since w is an RDP, $X'$ is again an RDP surface, and it follows from Proposition 2.3 that $\omega $ extends to $\omega '$ on $(X')^{\mathrm {sm}}$ , which generates $H^0(\operatorname {\mathrm {Spec}} \mathcal {O}_{X',w'} \setminus \{w'\}, \Omega ^2)$ at any closed point $w' \in X'$ above w, and that $D'(\omega ') = 0$ . Hence, $D'$ is symplectic at every fixed point above w. Since as above such fixed points are isolated, $Y'$ is smooth outside finitely many isolated points. Applying Proposition 2.10 to $\omega $ on $X \setminus \{w\}$ and $\omega '$ on $X' \setminus (\operatorname {\mathrm {Sing}}(X') \cup \operatorname {\mathrm {Fix}}(D'))$ , we obtain $2$ -forms $\psi $ on $Y \setminus \{\pi (w)\}$ and $\psi '$ on $Y' \setminus \pi ((\operatorname {\mathrm {Sing}}(X') \cup \operatorname {\mathrm {Fix}}(D')))$ , which are nonvanishing. Comparing $\psi $ and $\psi '$ , we observe that g is crepant.

Proof of Theorem 4.6(2)

By Remark 4.3 and Lemma 4.8(1), w is an isolated fixed point. By shrinking X, we may assume that D has no fixed point except w.

We construct a finite sequence $(X_j,D_j)_{0 \leq j \leq n}$ ( $n \geq 0$ ) of RDP surfaces $X_j$ and derivations $D_j$ on $X_j$ of multiplicative type that is symplectic at each fixed point. Let $(X_0,D_0) = (X,D)$ . If $X_j$ has no fixed RDP, then we terminate the sequence at $n = j$ . If $X_j$ has at least one fixed RDP, let $X_{j+1}$ be the blowup of $X_j$ at the fixed RDPs and $D_{j+1}$ the extension of $D_j$ to $X_{j+1}$ . Since any RDP becomes smooth after a finite number of blowups at RDPs, this sequence terminates at some $n \geq 0$ . By Lemma 4.8(2), $D_{j+1}$ on $X_{j+1}$ is symplectic at each fixed point, and $(X_{j+1})^{D_{j+1}} \to (X_j)^{D_j}$ is crepant. By Theorem 4.6(1) and Lemma 4.8(1), $Y_n = (X_n)^{D_n}$ has canonical singularity (i.e., has no singularity other than RDPs), and since $Y_n \to Y = X^D$ is crepant, also Y has canonical singularity. If $n> 0$ , then $\pi (w)$ is not a smooth point since $Y_n \to Y$ is a crepant morphism non-isomorphic at that point, and if $n = 0$ , then $\pi (w)$ is not a smooth point by Lemma 4.8(1). Hence, in either case, $\pi (w)$ is an RDP.

Moreover, we can classify all possible symplectic $\mu _p$ -actions on RDPs.

Proposition 4.9. Assume w is a fixed RDP, and the action is symplectic at w. Then there is a $\mu _p$ -equivariant isomorphism $\hat {\mathcal {O}}_{X,w} \cong k[[x,y,z]]/(F)$ with F equal to one in Table 4 and $\mu _p$ acts on $x,y,z$ by respective weights $a,-a,0$ for some $a \in \mathbb F_p^*$ . The singularities of X, $X' = \operatorname {\mathrm {Bl}}_w X$ , $X/\mu _p$ , and $X'/\mu _p$ are displayed in the table.

Table 4 Symplectic $\mu _p$ -actions on RDPs.

$\bullet \ A_0$ is a smooth point that is an isolated fixed point of D.

$\bullet \ [\mathsf {n}]$ means that the RDP is not fixed by D.

$\bullet \ \lfloor q \rfloor := \max \{n \in \mathbb Z \mid n \leq q\}$ denotes the integer part of a real q.

$\bullet \ q^+ := \max \{q,0\}$ denotes the positive part of a real q.

$\bullet \ [*]$ : It follows from the classification that for each (formal) isomorphism class of RDP, there

exists only one fixed symplectic $\mu _p$ -action up to isomorphism, except for the case of $D_{2n}^{n-1}$ ( $n \geq 3$ )

in $p = 2$ , in which case there are two and they are distinguished by the degree $2$ part $F_2$ being a

square of a homogeneous element or not. We distinguish them by notation $D_{2n}^{n-1}$ and $D_{2n}^{n-1}[*]$ . We

use the convention that $D_4^{1}[*] = D_4^{1}$ .

Remark 4.10. A polynomial $f \in k[x_1, \dots , x_m]$ is called quasi-homogeneous if, for some $a_1, \dots , a_m \in \mathbb Z_{\geq 1}$ , the monomials appearing in f have the same degree with respect to a (i.e., degree of the monomial $x_1^{i_1} \dots x_m^{i_m}$ is $i_1 a_1 + \dots + i_m a_m$ ). RDPs whose completions are not defined by quasi-homogeneous polynomials, which exist only if $p = 2,3,5$ , are precisely $D_n^r$ and $E_n^r$ with $r \neq 0$ . It follows from the classification given in Proposition 4.9 (resp. given in Proposition 4.7, resp. which is omitted) that if an RDP of type $D_n$ or $E_n$ admits a fixed symplectic (resp. non-fixed, resp. fixed non-symplectic) $\mu _p$ -action, then the singularity is not defined (resp. is defined, resp. is defined) by a quasi-homogeneous polynomial. We do not know any explanation of this phenomenon.

Proof of Proposition 4.9

We consider tuples $(a,b,c,F)$ with $a,b,c \in \mathbb F_p$ , not all $0$ , and $F \in k[[x,y,z]]$ such that $F = 0$ defines an RDP and only monomials of weight $a+b+c$ ( $\in \mathbb F_p$ ) appear in F, where $x,y,z$ have respective weights $a,b,c$ . By Lemma 4.5(2), it suffices to consider $k[[x,y,z]]/(F)$ of this form. We show that there exist a $\mu _p$ -equivariant isomorphism $k[[x,y,z]]/(F) \cong k[[x',y',z']]/(F')$ with $F'$ in Table 4 and $\operatorname {\mathrm {wt}}(x',y',z') = (1,-1,0)$ up to $\operatorname {\mathrm {Aut}}(\mu _p) = \mathbb F_p^*$ (which amounts to replacing $(a,b,c)$ with $(ta,tb,tc)$ for some $t \in \mathbb F_p^*$ ).

We write $F = \sum _{h,i,j} a_{hij} x^h y^i z^j$ , and we say that a polynomial or a formal power series has a monomial if its coefficient is nonzero.

First, assume the degree $2$ part $F_2$ is a non-square. Then we may assume that $F_2$ contains a non-square monomial, say $xy$ . (Indeed, if this is not the case, then $p \neq 2$ and $F_2$ contains at least two square monomials, say $x^2$ and $y^2$ , then x and y has the same weight, and then after a linear coordinate change, we may assume $F_2$ contains $xy$ .) Then we have $c = 0$ . If $a + b \not \in \{0,a,b\}$ , then $F \in (x,y)^2$ , which implies that $F = 0$ is not an RDP. If $a + b = a \neq 0$ , then $F \in (x)$ , again not an RDP. The same if $a + b = b \neq 0$ . So we have $a + b = 0$ and hence $F \in k[[x^p,xy,y^p,z]]$ . Since F cannot belong to $(x,y)$ , there exists an integer m such that F has the monomial $z^{m}$ . Let m be the smallest such integer. We have $F = u_1 z^m + u_2 xy + g_1(x^p) + g_2(y^p)$ for some units $u_1,u_2 \in k[[x^p, x y, y^p, z]]^*$ and power series $g_1, g_2$ . We may assume $u_1,u_2 \equiv 1 \pmod {\mathfrak {m}}$ . We eliminate $g_1, g_2$ by replacing x with $x + g_2/y$ and y with $y + g_1/x$ (and repeating this), and we obtain $F = u_1 z^m + u_2 xy$ . By replacing $x,y,z,F$ with suitable multiples, we obtain $F = z^m + xy$ .

Next, assume $p \geq 3$ and $F_2$ is square. We may assume $F_2 = z^2$ . We may assume $F_3 \not \equiv 0 \pmod {z}$ . If $F_3$ has $x^2y$ , then by $2c = 2a + b = a + b + c$ , we have $b = 0$ and $a = c$ , and hence $F \in (x,z)^2$ , which is absurd. Hence, we may assume $F_3$ has $y^3$ . By $2c = 3b = a+b+c$ , we have $(a,b,c) = (a,2a,3a)$ . If F does not have $x^3$ , then $F \in (z^2, x^3z, xyz, x^6, x^4y, x^2y^2, y^3)$ , and $F = 0$ cannot define an RDP. Hence, F has $x^3$ , hence $p = 3$ , and then $F \in k[[x^3,xy,y^3,z]]$ . We may assume that F does not have $xyz$ . To define an RDP, F must have one of $x^2y^2, x^4y, xy^4$ .

If it has $x^2y^2$ , then it is $E_6^1$ . We can eliminate $x^h y^i z$ , and we have

$$ \begin{align*} F &= z^2 u_1 + x^3 + y^3 + x^2 y^2 u_2 + \sum_{(h,i,j) \in S_1} a_{hij} x^h y^i z^j + \sum_{(h,i) \in S_2} b_{hi} x^h y^i + \sum_{(h,i) \in S_3} c_{hi} x^h y^i, \\ &\qquad S_1 = \{(4,1,0), (1,4,0)\}, \qquad S_2 = \{(6,0), (7,1)\}, \qquad S_3 = \{(0,6), (1,7)\}, \end{align*} $$

where $a_{hij} \in k$ , $b_{hi} \in k[[x^3]]$ , $c_{hi} \in k[[y^3]]$ , and $u_1, u_2 \in k[[x^3, x y, y^3, z]]^*$ . By replacing x with $x + t y^2$ and y with $y + t' x^2$ , we eliminate $a_{410}$ and $a_{140}$ . Then, by replacing F with $(1 + x^3 b_{60} + x^4 y b_{71} + y^3 c_{06} + x y^4 c_{17}) F$ , we eliminate all $b_{hi}$ and $c_{hi}$ . Finally, we replace $x,y,z,F$ with suitable multiples and achieve $u_1 = u_2 = 1$ . (For example, we let $F = u_2^{-3} F'$ , $x = u_2^{-1} x'$ , $y = u_2^{-1} y'$ , and $z = (u_1 u_2^3)^{-1/2} z'$ .)

If it does not have $x^2 y^2$ but has $x^4 y$ or $x y^4$ , then it is $E_8^1$ . By replacing F with a unit multiple, we may assume that it has $x^4 y$ and does not have $x y^4$ . We can eliminate $x^h y^i z$ , and we have

$$ \begin{align*} &F = z^2 u_1 + x^3 + y^3 + x^4 y u_2 + \sum_{(h,i,j) \in S_1} a_{hij} x^h y^i z^j + \sum_{(h,i) \in S_2} b_{hi} x^h y^i + \sum_{(h,i) \in S_3} c_{hi} x^h y^i, \\ S_1 &= \{(6,0,0), (3,3,0), (0,6,0)\}, \qquad S_2 = \{(9,0)\}, \qquad S_3 = \{(0,9), (1,7), (2,5), (3,6)\}, \end{align*} $$

with $a_{hij}$ , $b_{hi}$ , $c_{hi}$ , and $u_1,u_2$ as in the previous case. By replacing y with $y + t x^2$ , we eliminate $a_{600}$ . By replacing x with $x + t' y^2$ and F with $(1 + t" y^3) F$ , we eliminate $a_{330}$ and $a_{060}$ . By replacing F with $(1 + x^2 y^2 c_{25} + x y^4 c_{17} + y^6 c_{09}) F$ , then with $(1 + x^6 b_{90} + x^3 y^3 c_{36}) F$ , we eliminate all $b_{hi}$ and $c_{hi}$ . We replace $x,y,z,F$ with suitable multiples and achieve $u_1 = u_2 = 1$ .

Hereafter, assume $p = 2$ and that $F_2$ is a square. If $\sqrt {F_2}$ is homogeneous, then we may assume $F_2 = x^2$ , we may assume $F_3$ has $y^2 z$ or $z^3$ , and then we have $(a,b,c) = (1,1,0)$ , and hence $F \in k[[x^2,xy,y^2,z]]$ . If $\sqrt {F_2}$ is not homogeneous, then we may assume $F_2 = x^2 + z^2$ and $a = 1$ and $c = 0$ , and then again we have $(a,b,c) = (1,1,0)$ , and hence $F \in k[[x^2,xy,y^2,z]]$ , and $F_3$ has $x y z$ , $y^2 z$ , $z^3$ , or $x^2 z$ .

Assume ( $F_2$ is $x^2$ or $x^2 + z^2$ and) $F_3$ contains $y^2 z$ . Furthermore, since $F \notin (x,y)^2$ , we see that F has $z^l$ ( $l \geq 2$ ). We have $F \equiv x^2 + y^2 z + z^l \pmod {(x^4, x^3 y, x^2 y^2, x y^3, y^4, x^2 z, x y z, y^2 z^2, z^{l+1})}$ . Write $F = F_0(x^2, y^2, z) + xy F_1(x^2, y^2, z)$ . Then there exist unique $f, g \in k[[z]]$ such that $F_0 \in (x^2 - f(z)^2, y^2 - g(z)^2)$ , and they satisfy $l = \min \{2 \operatorname {\mathrm {ord}}_z(f), 2 \operatorname {\mathrm {ord}}_z(g) + 1\}$ . If l is even, then, by replacing y with $y - x g/f$ , we may assume $g = 0$ . If l is odd, then, by replacing x with $x - y f/g$ , we may assume $f = 0$ . We eliminate $a_{hij}$ with $h \geq 2$ , $(h,i,j) \neq (2,0,0)$ , by replacing F with $(1 + a_{hij} x^{h-2} y^i z^j) F$ , and $a_{hij}$ with $i \geq 2$ , $(h,i,j) \neq (0,2,0), (0,2,1)$ , by replacing z with $z + a_{hij} x^h y^{i-2} z^j$ . We obtain $F = x^2 + y^2 z + z^l u(z) + x y e(z)$ , where $e(z) \in k[[z]]$ and $u(z) \in k[[z]]^*$ . We have $e(z) \neq 0$ , since if $e(z) = 0$ , then $F = F_0 \in ((x - f(z))^2, (y - g(z))^2)$ , which is absurd. Write $e(z) = z^k v(z)$ , $k \geq 1$ , and $v(z) \in k[[z]]^*$ . It is $D_{2k+l}^{\lfloor l/2 \rfloor }$ . If l is even, then, since $F_0 \in (x^2 - f(z)^2, y^2 - g(z)^2)$ and $g = 0$ , we have $z^l u(z) = f(z)^2$ and hence $u(z)$ is a square, and then by replacing x with $u(z)^{1/2} x$ and by replacing F with a unit multiple, we obtain $F = x^2 + y^2 z u'(z) + z^l + x y e(z)$ for some $u'(z) \in k[[z]]^*$ . Similarly, if l is odd, then, since $f = 0$ , we have $z^l u(z) = z g(z)^2$ and hence $u(z)$ is a square, and then (by replacing y) we obtain $F = x^2 u'(z) + y^2 z + z^l + x y e(z)$ . By replacing $x,y,z,F$ with unit multiples, we can achieve $u' = v = 1$ .

Assume $F_2 = x^2$ and $F_3$ has $z^3$ but no $y^2z$ . To define an RDP, F must have $y^4$ and must have $xyz$ or $xy^3$ . If F has $xyz$ , then it is $E_7^3$ . We have

$$ \begin{align*} &\qquad \quad F = x^2 + y^4 + z^3 u_1+ x y z u_2 + \sum_{(h,i,j) \in S_1} a_{hij} x^h y^i z^j \\ &\qquad \quad + \sum_{(h,i) \in S_2} b_{hi} x^h y^i + \sum_{(h,i) \in S_3} c_{hi} x^h y^i + \sum_{(h,i) \in S_4} d_{hi} x^h y^i, \\ S_1 &= \{(3,1,0), (1,3,0), (1,5,0), (0,2,2), (2,0,1), (2,0,2), (0,4,1), (0,4,2)\}, \\ &\quad S_2 = \{(4,0), (5,1)\}, \qquad S_3 = \{(0,6), (1,7)\}, \qquad S_4 = \{(2,2)\}, \end{align*} $$

where $a_{hij} \in k$ , $b_{hi} = \sum _{j = 0}^{2} b_{hij} z^j$ with $b_{hij} \in k[[x^2]]$ , $c_{hi} = \sum _{j = 0}^{2} c_{hij} z^j$ with $c_{hij} \in k[[y^2]]$ , $d_{hi} \in k[[x^2, x y, y^2]]$ , and $u_1,u_2 \in k[[x^2, x y, y^2, z]]^*$ . We may assume $u_1, u_2 \equiv 1 \pmod {\mathfrak {m}}$ . We replace z with $z + a_{310} x^2 + a_{130} y^2 + a_{150} y^4$ , x with $x + t y z$ (which eliminates $a_{022}$ ), y with $y + a_{201} x + a_{202} x z$ , and x with $x + a_{041} y^3 + a_{042} y^3 z$ , and thus eliminate all $a_{hij}$ ( $(h,i,j) \in S_1$ ). We replace F with $(1 + x^2 b_{40} + y^2 c_{06}) F$ , F with $(1 + x^3 y b_{51} + x y^3 c_{17}) F$ , and z with $z + d_{22} x y$ , and thus eliminate all $b_{hi}$ , $c_{hi}$ , and $d_{hi}$ . We replace $x,y,z,F$ with suitable multiples and achieve $u_1= u_2 = 1$ .

Next, if F does not have $xyz$ but has $x y^3$ , then it is $E_8^3$ . We have

$$ \begin{align*} &F = x^2 + y^4 + z^3 u_1 + x y^3 u_2 + \sum_{(h,i,j) \in S_1} a_{hij} x^h y^i z^j + \sum_{(h,i) \in S_2} b_{hi} x^h y^i + \sum_{(h,i) \in S_3} c_{hi} x^h y^i, \\ &\quad S_1 = \{(2,0,1), (2,0,2), (0,4,1), (0,2,2), (1,1,2), (0,6,0), (0,4,2), (0,6,1), (0,6,2), \\ &\qquad (2,2,0), (2,2,1), (2,2,2)\}, \qquad S_2 = \{(4,0), (3,1), (4,2))\}, \qquad S_3 = \{(0,8)\}, \end{align*} $$

with $a_{hij}$ , $b_{hi}$ , $c_{hi}$ , and $u_1,u_2$ as in the previous case. We may assume $u_1, u_2 \equiv 1 \pmod {\mathfrak {m}}$ . We replace F with $(1 + a_{201} z + a_{202} z^2) F$ , x with $x + t y z$ and z with $z + t' y^2$ (which eliminates $a_{041}$ and $a_{022}$ ), z with $z + a_{112} x y$ , x with $x + t" y^3$ (which eliminates $a_{060}$ ), x with $x + a_{042} y z^2 + a_{061} y^3 z$ , x with $x + a_{062} y^3 z^2$ , and y with $y + (a_{220} + a_{221} z + a_{222} z^2) x$ , and thus eliminate all $a_{hij}$ ( $(h,i,j) \in S_1$ ). We replace F with $(1 + x^2 b_{40} + x y b_{31} + x^2 y^2 b_{42} + y^4 c_{08}) F$ , and thus eliminate all $b_{hi}$ and $c_{hi}$ . We replace $x,y,z,F$ with suitable multiples and achieve $u_1 = u_2 = 1$ .

Assume $F_2 = x^2 + z^2$ and $F_3$ does not have $y^2 z$ and has $xyz$ . F moreover needs $x y^i$ , $y^i z$ , or $y^l$ . Replacing z with $z + xy^{i-1}$ (resp. x with $x + y^{i-1}z$ ), we may assume that there are no $x y^i$ (resp. $y^i z$ ) of low degree. Thus, we have

$$ \begin{align*} &\qquad\qquad\qquad\qquad\quad F \equiv x^2 + z^2 + xyz + y^{2n-2} \\ &\pmod{(x^4, x^3 y, x^2 y^2, x y^{m} (m> n), y^{2n}, x^2 z, x y z^2, y^2 z^2, y^{m} z (m > n), z^3)}, \end{align*} $$

$n \geq 3$ , and this is $D_{2n}^{n-1}[*]$ . We eliminate $x^h y^i$ ( $h ,i \geq 1$ , $(h, i) \neq (1, 1)$ ) by replacing z with $z + a_{hi0} x^{h-1} y^{i-1}$ , and $x^h z^j$ and $y^i z^j$ similarly. We eliminate $x^h$ , $y^i$ , $z^j$ ( $h \geq 3$ , $i \geq 2n-1$ , $j \geq 3$ ) by replacing F with a unit multiple. We obtain $F = x^2 + z^2 + xyz u + y^{2n-2}$ for some $u \in k[[x^2, x y, y^2, z]]^*$ , and we can achieve $u = 1$ .

Assume $F_2 = x^2 + z^2$ and $F_3$ does not have $y^2 z$ nor $xyz$ . By replacing F with $(1 + a_{201} x^2 z)^{-1} F$ , we may assume that F does not have $x^2 z$ . Then F has $z^3$ and F moreover needs $xy^3$ , and then it is $E_7^2$ . We have

$$ \begin{align*} &F = x^2 + z^2 + z^3 u_1 + x y^3 u_2 + \sum_{(h,i,j) \in S_1} a_{hij} x^h y^i z^j + \sum_{(h,i) \in S_2} b_{hi} x^h y^i + \sum_{(h,i) \in S_3} c_{hi} x^h y^i, \\ & \ \ S_1 = \{(0,4,0), (0,4,1), (0,6,0), (0,2,2), (3,1,0), (2,2,0), (1,1,2), (4,0,0), (2,0,2) \}, \\ &\qquad\qquad\qquad\qquad S_2 = \{(4,0), (3,1), (2,2)\}, \qquad S_3 = \{(0,4)\}, \end{align*} $$

with $a_{hij}$ , $b_{hi}$ , $c_{hi}$ , and $u_1,u_2$ as in the case of $E_7^3$ , and moreover $b_{hi} \in (x^2, z)$ and $c_{hi} \in (y^4, y^2 z, z^2)$ . We may assume $u_1, u_2 \equiv 1 \pmod {\mathfrak {m}}$ . We replace z with $z + a_{040}^{1/2} y^2$ , x with $x + a_{041} y z$ , x with $x + t y^3$ (which eliminates $a_{060}$ ), F with $(1 + a_{022} y^2) F$ , y with $y + t' x$ and z with $z + t" x y$ and F with $(1 + t"' x y) F$ (which eliminates $a_{310}$ , $a_{220}$ , and $a_{112}$ ), z with $z + t"" x^2$ and F with $(1 + t""' x^2) F$ (which eliminates $a_{400}$ and $a_{202}$ ), and thus eliminate all $a_{hij}$ ( $(h,i,j) \in S_1$ ). We replace x with $x + y c_{04}$ and F with $(1 + b_{40} x^2 + b_{31} x y + b_{22} y^2) F$ to eliminate all $b_{hi}$ and $c_{hi}$ . We replace $x,y,z,F$ with suitable multiples and achieve $u_1 = u_2 = 1$ .

4.3 $\mu _n$ -actions on RDPs

In this section, we classify $\mu _n$ -actions on RDPs under each of the following assumptions.

• • w is not fixed by $\mu _n$ (Proposition 4.12).

• • $n = p^e$ , and the subgroup scheme $\mu _p$ fixes w and is symplectic (Proposition 4.13).

• • w is fixed by $\mu _n$ and the action is symplectic (Proposition 4.14).

In Propositions 4.13 and 4.14, we use the convention that a smooth point is of type $A_0$ .

Lemma 4.11. Let X be a k-scheme equipped with a $\mu _{p^2}$ -action. Let $\pi _1 \colon X \to X_1 = X / \mu _p$ be the quotient morphism by the action of the subgroup scheme $\mu _p \subset \mu _{p^2}$ . If $w \in X$ is non-fixed by the action of $\mu _p$ , then $\pi _1(w) \in X_1$ is non-fixed by the action of $\mu _{p^2}/\mu _p$ .

Proof. Let $\mathcal {O}_{X,w} = B = \bigoplus _{i \in \mathbb Z/p^2\mathbb Z} B_i$ be the corresponding decomposition. Since w is non-fixed by $\mu _p$ , there exists $y \in \mathfrak {m}_{w} \subset B$ with $1 + y \in \bigoplus _{i \equiv 1 \pmod {p}} B_i$ (Lemma 2.9). Then, since $y^p \in \mathfrak {m}_{\pi _1(w)} \subset \mathcal {O}_{X_1, \pi _1(w)}$ satisfies $1 + y^p = (1 + y)^p \in B_p$ , we conclude by Proposition 2.8 that $\pi _1(w)$ is non-fixed by $\mu _{p^2}/\mu _p$ .

Suppose X is a scheme equipped with a $\mu _n$ -action, $n = p^e r$ with $p \nmid r$ , and $w \in X$ is a closed point fixed by $\mu _r \subset \mu _n$ . Let f be the maximal integer with $0 \leq f \leq e$ such that the subgroup scheme $\mu _{p^f} \subset \mu _{p^e} \subset \mu _n$ fixes w. We say that $\mu _{p^f r}$ is the stabilizer of w and denote it by $\operatorname {\mathrm {Stab}}(w)$ .

Proposition 4.12. Let $\mathcal {O}_{X,w}$ , together with a $\mu _n$ -action, be as in the beginning of §4, and assume w is an RDP. Write $n = p^e r$ with $p \nmid r$ , and $\operatorname {\mathrm {Stab}}(w) = \mu _{p^f r}$ . Suppose $\operatorname {\mathrm {Stab}}(w) \subsetneq \mu _n$ (hence $f < e$ , in particular $e \geq 1$ ). Then there exist $x,y,z \in \mathfrak {m}$ generating $\mathfrak {m}$ , with $x,y,1+z$ homogeneous such that, up to replacing r with a multiple and up to $\operatorname {\mathrm {Aut}}(\mu _n) = (\mathbb Z/n\mathbb Z)^*$ , the weights and the type of singularity are as in Table 5.

Table 5 Non-fixed $\mu _n$ -actions on RDPs (listed in order of appearance in the proof).

(*): In the cases where $n = 4(m-1)$ or $n = 4m$ , $p^e$ is the highest power of $p = 2$ dividing n.

In this case, we do not pursue the exact equation, and the equations in Table 5 are merely examples.

Proof. We first show that there exist elements $x,y,z \in \mathfrak {m}$ generating $\mathfrak {m}$ and satisfying $\operatorname {\mathrm {wt}}(x,y,1+z) = (a,b,c)$ with $c = n/p^{e-f} = p^f r$ . Since the stabilizer of the $\mu _{n}$ -action is $\mu _{p^f r} \subsetneq \mu _{n }$ , there exist $i \in \mathbb Z/n\mathbb Z$ and a homogeneous element t of weight i such that $t \notin \mathfrak {m}$ and $\gcd \{i, n\} = p^f r$ . We may assume $t \equiv 1 \pmod {\mathfrak {m}}$ , and then $z' := -1 + t \in \mathfrak {m}$ satisfies $\operatorname {\mathrm {wt}}(1 + z') = i$ . Take an integer q such that $q i = p^f r$ (in $\mathbb Z/n\mathbb Z$ ), and then $1 + z := (1 + z')^q$ satisfies $z \in \mathfrak {m}$ and $\operatorname {\mathrm {wt}}(1 + z) = p^f r$ . Now, take $x^{\prime (1)}, x^{\prime (2)}, x^{\prime (3)}$ generating $\mathfrak {m}$ . We may assume each $x^{\prime (h)}$ is homogeneous with respect to $\mu _{p^f r}$ , that is, there exists $i^{(1)}, i^{(2)}, i^{(3)} \in \mathbb Z/n\mathbb Z$ such that $x^{\prime (h)} = \sum _{j = 0}^{p^{e-f}-1} x^{\prime (h,j)}$ with $\operatorname {\mathrm {wt}}(x^{\prime (h,j)}) = i^{(h)} + j p^f r$ . Let $x^{(h)} := \sum _{j = 0}^{p^{e-f}-1} (1+z)^{-j} x^{\prime (h,j)}$ . Then $\operatorname {\mathrm {wt}}(x^{(h)}) = i^{(h)}$ and, since $x^{(h)} \equiv x^{\prime (h)} \pmod {(z)}$ , the elements $x^{(1)}, x^{(2)}, x^{(3)}, z$ generate $\mathfrak {m}$ . We can omit one of these four elements, which cannot be z since $\mathfrak {m}$ is not generated by homogeneous elements.

In this proof, by a monomial, we mean a polynomial of the form $x^i y^j z^{p^{e-f} l} (1+z)^m$ with $0 \leq m < p^{e-f}$ . Any polynomial (resp. formal power series) is uniquely expressed as a finite (resp. possibly infinite) sum of monomials with k-coefficients, and we say that a polynomial or a formal power series has a monomial if its coefficient is nonzero. Expressions such as $F = x^2 + y^3 (1+z) + \cdots $ will indicate that F has these monomials. However, when we say of degree m part $F_m$ of F, this is understood with respect to the usual monomials $x^i y^j z^l$ .

Assume the degree $2$ part $F_2$ of F is either irreducible or the product of two distinct homogeneous linear factors. Then we may assume $F = xy (1+z)^i + \cdots $ , and F must have $z^{m'} (1+z)^j$ , we may assume $i = j = 0$ by replacing y and F, and it is $A_{m'-1}$ with $m' = p^{e-f} m$ .

Assume $F_2$ is the product of two distinct nonhomogeneous linear factors. Then we have $p \neq 2$ and we may assume $F = x^2 (1+z)^i + y^2 (1+z)^j + z^{m'} (1+z)^k + \cdots $ and $b \equiv a + r/2 \pmod {r}$ . We may assume $k = 0$ by replacing F with $(1+z)^{-k} F$ and $i = j = 0$ by replacing x and y with $x (1+z)^{i(p^e+1)/2}$ and $y (1+z)^{j(p^e+1)/2}$ . We have $2a = 2b = 0$ and then we have $f = 0$ and $r = 2$ . (Otherwise, $a,b,c$ cannot generate $\mathbb Z/p^e r \mathbb Z$ .) It is $A_{m'-1}$ with $m' = p^e m$ , and we may assume $(a,b) = (0,n/2)$ .

Assume $p \geq 5$ and $F_2$ is a square. We may assume $F_2 = x^2$ , $F_3 = y^3$ , $F = x^2 + y^3 + \cdots $ , and then F must have $z^5 (1+z)^i$ and we have $p = 5$ , it is $E_8^0$ , and we may assume $i = 0$ . By $2a = 3b = 0$ , we have $n \mid 30$ and we may assume $a = 15$ and $b = 10$ .

Assume $p = 3$ and $F_2$ is a square. We may assume $F = x^2 + \cdots $ . We may assume $F_3 \bmod (x)$ is either $y^3$ , $z^3$ , or $y^3 + z^3$ . If $F = x^2 + y^3 + \cdots $ , then F must have $y z^3$ and it is $E_7^0$ . If $F = x^2 + z^3 + \cdots $ , then F must have $y^4$ or $y^5$ and it is $E_6^0$ or $E_8^0$ . If $F = x^2 + y^3 + z^3 (1+z)^i + \cdots $ , then we may assume $i = 1$ (if $i = 0$ , then by replacing z with $y (1 + z)^l + z $ , we reduce this case to the previous case) and then it is $E_6^0$ .

Assume $p = 2$ and $F_2$ is a square. We may assume $F_2$ is $x^2$ , $z^2$ , $x^2 + z^2$ , $x^2 + y^2$ , or $x^2 + y^2 + z^2$ .

Assume $F_2 = x^2$ . We may assume $F = x^2 + \cdots $ . Then F must have $y z^2 (1+z)^i$ or $y^3 (1+z)^j$ . If F has $y z^2 (1+z)^i$ , then we may assume $i = 0$ and F must have $x y^m (1+z)^i$ ( $m \geq 2$ ) or $y^m (1+z)$ ( $m \geq 3$ ). In the former case, we may assume $i = 0$ (by replacing x with $x (1+z)^i$ ) and we have $D_{2m+1}^0$ with $(a,b,c) = (2m,2,2m-1)$ , $n \mid 2(2m-1)$ . In the latter case, we have $D_{2m}^0$ with $(a,b,c) = (1,2,2(m-1))$ , $n \mid 4(m-1)$ . Now, assume that F does not have $y z^2 (1+z)^i$ and has $y^3 (1+z)^j$ . We may assume $j = 0$ . Then F must have either $x z^2 (1+z)^i$ or $z^4 (1+z)^i$ . If F has $x z^2 (1+z)^i$ , then we may assume $i = 0$ and then $F = x^2 + y^3 + x z^2 + \cdots $ is $E_6^0$ , and we have $e = 1$ , $f = 0$ , $a = 0$ , and $c = n/2$ . If F does not have $x z^2 (1+z)^i$ and has $z^4 (1+z)^i$ , then we may assume $i = 1$ and then $F = x^2 + y^3 + z^4 (1+z) + \cdots $ is $E_8^0$ , and we have $e - f \leq 2$ , $f = 1$ , $(a,b,c) \equiv (1,2/3,2) \pmod {2^e}$ .

Assume $F_2 = z^2$ . We have $e - f = 1$ . Suppose F has $x^2 y (1 + z)^i$ . We may assume $i = 0$ . We may assume F does not have monomials $z^2 M$ ( $M \neq 1$ ) nor $x^2 M$ ( $M \neq 1, y$ ) of low degree (by replacing F or y with $(1 + M)^{-1} F$ or $y + M$ , respectively). F must have $x y^m$ or $y^m (1+z)$ and then it is $D_{2m}^0$ or $D_{2m+1}^0$ . Now, suppose F does not have $x^2 y (1 + z)^i$ nor $x y^2 (1 + z)^i$ and has $x^3 (1 + z)^i$ . We may assume $i = 0$ . Then F must have $y^3 (1 + z)^j$ ( $D_4^0$ ), $x y^3 (1 + z)^j$ ( $E_7^0$ ), or $y^5 (1 + z)^j$ ( $E_8^0$ ).

Assume $F_2 = x^2 + z^2$ . We may assume $F = x^2 + z^2 (1+z)^i + \cdots $ . If $i = 0$ , then by replacing z with $z + x$ or $z + x (1+z)$ , we reduce this case to the previous case. Assume $i = 1$ . F cannot have $x^3 (1+z)^i$ nor $x z^2 (1+z)^i$ . (If F has $x^3 (1+z)^i$ , then we have $2a = c = 3a + ic$ and this implies $(2i+1)c = 0$ , contradicting $c = n/p^{e-f}$ . Other cases are similar.) We may assume that F does not have $y z^2 (1+z)^i$ . If F has $y^3 (1+z)^i$ , then F does not have $x y^2 (1+z)^j$ and it is $D_4^0$ . If F does not have $y^3 (1+z)^i$ and has $x^2 y (1+z)^i$ , then F cannot have $x y^2 (1+z)^j$ , and F must have $y^{2m+1} (1+z)^j$ , and it is $D_{4m}^0$ . If F does not have $y^3 (1+z)^i$ nor $x^2 y (1+z)^i$ and has $xy^2 (1+z)^i$ , then it is $D_5^0$ . If F does not have $y^3 (1+z)^i$ nor $xy^2 (1+z)^i$ nor $x^2 y (1+z)^i$ , then F must have $xy^3 (1+z)^i$ or $y^5 (1+z)^i$ , and (we may assume $i = 0$ and) it is $E_7^2$ or $E_8^0$ . (This $E_7^2$ is the only example of $D_n^r$ or $E_n^r$ with $r> 0$ in this proposition.)

Assume $F_2 = x^2 + y^2$ . Write $F = x^2 + y^2 (1+z)^j + \cdots $ . If $j = 0$ , then by replacing x with $x + y (1+z)^k$ , we reduce to the $F_2 = x^2$ case. Suppose $j = 1$ . We may assume F does not have monomials $x^2 M$ ( $M \neq (1+z)^i$ ) nor $y^2 M$ ( $M \neq (1+z)^i$ ) of low degree (by replacing F or z with $(1 + M)^{-1} F$ or $z + M$ , respectively). F must have $x z^{2m} (1+z)^k$ or $y z^{2m} (1+z)^k$ , by symmetry, we may assume F has $x z^{2m} (1+z)^k$ , we may assume $k = 0$ , and then it is $D_{4m}^0$ .

Assume $F_2 = x^2 + y^2 + z^2$ . Write $F = x^2 (1+z)^i + y^2 (1+z)^j + z^2 (1+z)^k + \cdots $ , $i,j,k \in \{0,1\}$ . If $i = j$ , then we reduce this case to $F = x^2 + z^2$ case by replacing x with $x + y (1+z)^l$ . If $i \neq j$ , then either $i = k$ or $j = k$ and then we reduce this case to $F = x^2 + z^2$ case by replacing z with $z + x (1+z)^l$ or $z + y (1+z)^l$ .

Proposition 4.13. Let $\mathcal {O}_{X,w}$ , together with a $\mu _n$ -action, be as in the beginning of §4. Suppose $n = p^e$ with $p> 0$ and $e \geq 2$ . Let $\operatorname {\mathrm {Stab}}(w) = \mu _{p^f}$ ( $0 \leq f \leq e$ ). Suppose $f> 0$ and that the subgroup scheme $\mu _p \subset \operatorname {\mathrm {Stab}}(w)$ acts symplectically. Then one of the following is true.

• • w is $A_{p^{e-f} m-1}$ for some integer $m \geq 1$ .

• • w is $E_7^2$ and $(p^f, p^e) = (2, 4)$ .

• • w is $D_{2n+1}^{n-1}$ ( $n \geq 2$ ) or $E_8^3$ , and $(p^f, p^e) = (4, 4)$ .

Proof. Let $\mathcal {O}_{x,w} = B = \bigoplus _{i \in \mathbb Z/p^2\mathbb Z} B_i$ be the corresponding decomposition.

Assume w is a smooth point. Since $\mu _p$ acts symplectically, the maximal ideal $\mathfrak {m}$ is generated by two elements $x \in \bigoplus _{i \equiv a \pmod {p}} B_i$ and $y \in \bigoplus _{i \equiv b \pmod {p}} B_i$ for some $a,b \in \mathbb Z/p^e\mathbb Z$ with $a,b \not \equiv 0$ and $a + b \equiv 0 \pmod {p}$ . Since $a,b \not \equiv 0 \pmod {p}$ , we may assume moreover $x \in B_a$ and $y \in B_b$ . Then w is fixed by the whole group scheme $\mu _{p^e}$ and hence $e = f$ . This case is done (with $m = 1$ : recall the convention that a smooth point is of type $A_0$ ).

Hereafter, we assume w is an RDP. Let $\varepsilon = 0$ if $e = f$ and $\varepsilon = 1$ if $e> f$ . By arguing as in the beginning of the proof of Proposition 4.12 and by using Proposition 4.9, $\mathfrak {m}$ is generated by three elements $x,y,z$ with $x \in B_a$ , $y \in B_b$ , and $\varepsilon + z \in B_c$ , and we may assume $a \equiv -b \not \equiv 0 \pmod {p}$ , and if $e> f$ then we may moreover assume $c = p^f$ .

If $e> f$ , then, since $\mu _p$ acts symplectically, it follows (from the classification given in Proposition 4.12) that either the RDP is $A_{m'-1}$ and then we may assume $F = xy + z^{m'} + \cdots $ and hence $p^{e-f} \mid m'$ , or the RDP is $E_7^2$ and $(p^f, p^e) = (2, 4)$ .

Hereafter, assume $e = f$ . If the RDP is $A_{m-1}$ , then there is nothing to prove. The remaining possibilities are given in Table 4 (Proposition 4.9) and in particular we have $p \leq 3$ .

Assume $p = 3$ and the RDP is $E_6^1$ or $E_8^1$ (as in Proposition 4.9). We may assume that $F = z^2 + x^3 + y^3 + \cdots $ with $\operatorname {\mathrm {wt}}(x,y,z) \equiv (1,-1,0) \pmod {3}$ . Then F cannot be homogeneous since $\operatorname {\mathrm {wt}}(x^3) \not \equiv \operatorname {\mathrm {wt}}(y^3) \pmod {3^2}$ .

Assume $p = 2$ and the RDP is $D_n$ or $E_n$ . By the classification in Proposition 4.9, we have $(a,b,c) \equiv (1,1,0) \pmod {2}$ , and $F_2 \not \in k z^2$ . If $(a,b,c) \equiv (1,\pm 1,0) \pmod {4}$ , then $F \in (x,y)^2$ and F cannot define an RDP. If $(a,b,c) \equiv (1,1,2) \pmod {4}$ (then we may assume $F_2 = x^2$ ), or if $(a,b,c) \equiv (1,-1,2) \pmod {4}$ and $F_2 = x^2$ , then $F \in (x^2,xyz,xy^3,z^3,z^2y^2,zy^4,y^6)$ and hence F cannot define an RDP. Hence, we may assume $(a,b,c) \equiv (1,-1,2) \pmod {4}$ and $F_2 = x^2 + y^2$ , and hence $p^e = 4$ . If F has $xyz$ , then F must have $z^{2n-1}$ for some $n \geq 2$ , and then it is $D_{2n+1}^{n-1}$ . If F does not have $xyz$ , then F must have $z^3$ and F must also have $x^3 y$ or $x y^3$ , and then it is $E_8^3$ .

Proposition 4.14. Let $p \geq 0$ . Let $\mathcal {O}_{X,w}$ , together with a $\mu _n$ -action ( $n> 1$ ), be as in the beginning of §4. Suppose the action fixes w and is symplectic. Then p, n, the type of singularity at w, and the quotient singularity are as in Table 6.

Table 6 Symplectic $\mu _n$ -actions on RDPs.

Proof. If $p \nmid n$ , then this is Remark 3.5 (Table 2). If $n = p$ , then this is Proposition 4.9 (Table 4). If $n = p^e$ with $e \geq 2$ , then by Proposition 4.13, the possibilities are $D_{2n+1}^{n-1}$ , $E_8^3$ , and $A_{m-1}$ (with quotient $A_{mn-1}$ ). In the other cases, we conclude by comparing the tables of the tame case and the $n = p$ case. For example, $E_6^r$ with $(p,n) = (3,6)$ is impossible since the $\mu _2$ -quotient $E_7^r$ of $E_6^r$ does not admit a symplectic $\mu _3$ -action.

To a point w fixed by a $\mu _n$ -action, we define its multiplicity $m(w)$ inductively by:

• • if w is a smooth point, then $m(w) = 1$ , and

• • if w is an RDP, then $m(w) = \sum _{w' \in \operatorname {\mathrm {Fix}}(\mu _n \curvearrowright \operatorname {\mathrm {Bl}}_w X)} m(w')$ .

The multiplicity for each case is displayed in Table 6. We observe that $m(\pi _r(w)) = r m(w)$ for any divisor $r < n$ of n, where ${\pi _r}:{X}\to {X/\mu _r}$ is the quotient map and $m(\pi _r(w))$ is the multiplicity with respect to the $\mu _{n/r}$ -action on $X/\mu _r$ .

4.4 $\mu _n$ -actions on smooth points

Let $B = \mathcal {O}_{X,w}$ , together with a $\mu _n$ -action, be as in the beginning of §4, and assume it is smooth. Assume the $\mu _n$ -action fixes w and is symplectic at w. As shown in Lemma 4.8(1), there exists $j \in (\mathbb Z/n\mathbb Z)^*$ , unique up to sign, such that the maximal ideal of B is generated by two homogeneous elements of respective weights j and $-j$ . (We say that the weights of the $\mu _n$ -action on the tangent space are j and $-j$ .)

Now, let $\tilde {Y}$ be the minimal resolution of $Y = X / \mu _n$ at $\pi (w)$ , and let $\pi ' \colon X' = X \times _Y \tilde {Y} \to \tilde {Y}$ . Let $e_{k}$ ( $k = 1, \dots , n-1$ ) be the exceptional curves of $\tilde {Y}$ , ordered in a way that $e_{k} \cap e_{k'} \neq \emptyset $ if and only if $\lvert k - k' \rvert \leq 1$ . The $\mu _n$ -action induces a decomposition $\pi ^{\prime }_* \mathcal {O}_{X'} = \bigoplus _{i \in \mathbb Z/n\mathbb Z} (\pi ^{\prime }_* \mathcal {O}_{X'})_i$ . Let $I_i := \operatorname {\mathrm {Im}}( ((\pi ^{\prime }_* \mathcal {O}_{X'})_i)^{\otimes n} \to \mathcal {O}_{\tilde {Y}})$ for $i = 0, \dots , n-1$ . Then $I_i$ are described as follows. (Clearly, $I_0 = \mathcal {O}_{X'}$ .)

Lemma 4.15. $(\pi ^{\prime }_* \mathcal {O}_{X'})_i$ and $I_i$ are invertible sheaves. After possibly reversing the ordering of the exceptional curves, we have an equality

$$\begin{align*}I_i = \mathcal{O}_{\tilde{Y}} \Bigl( -\sum_k f_n \bigl( (j^{-1} i \bmod n), k \bigr) e_k \Bigr), \end{align*}$$

for each $i = 1, \dots , n-1$ .

Here, $j^{-1} i \bmod n$ denotes the unique integer $h \in \{0, 1, \dots , n-1\}$ satisfying $h j \equiv i \pmod {n}$ , and the function ${f_n}:{\{1, 2, \dots , n-1\}^2}\to {\mathbb Z}$ is defined as

$$\begin{align*}f_n(h, k) = \begin{cases} h k, & (k \leq n - h), \\ (n - h) (n - k), & (k \geq n - h). \end{cases} \end{align*}$$

Proof. Straightforward.

5 Tame quotients of K3 surfaces and abelian surfaces

The following fact should be known to experts. (For example, if X is a K3 surface in characteristic $0$ and G is symplectic and commutative, then this is a result of Nikulin.) We give a proof since we could not find a complete reference (covering all characteristics).

Theorem 5.1. Let X be either an abelian surface or an RDP K3 surface in characteristic $p \geq 0$ , and G a finite group of order not divisible by p acting on X.

If X is an RDP K3 surface and G is symplectic, then the quotient $X/G$ is an RDP K3 surface.

If X is an abelian surface and G is symplectic, then $X/G$ is either an abelian surface or an RDP K3 surface.

If X is an RDP K3 surface and G is non-symplectic, then $X/G$ is either an RDP Enriques surface or a rational surface.

If X is an abelian surface and G is non-symplectic, then $X/G$ is either an RDP Enriques surface, a (quasi-)hyperelliptic surface, a surface birational to a ruled surface, or a rational surface.

Proof. By Proposition 3.4, we may assume X is smooth. Let ${\pi }:{X}\to {X/G = Y}$ be the quotient morphism, and let $\tilde {Y} \to Y$ be the minimal resolution.

We have $b_1(\tilde {Y}) = b_1(Y) \leq b_1(X)$ , where $b_i(X) = \dim H_{\mathrm {\acute et}}^i(X, \mathbb Q_l)$ is the ith l-adic Betti number for an auxiliary prime $l \neq \operatorname {\mathrm {char}} k$ . Hence, if X is a K3 surface (hence $b_1(X) = 0$ ), then $\tilde {Y}$ cannot be abelian, (quasi-)hyperelliptic, nor nonrational ruled.

First, suppose G acts non-symplectically. If some nontrivial $g \in G$ satisfy $\dim \operatorname {\mathrm {Fix}}(g) = 1$ , then by the usual ramification formula, $K_Y$ has negative coefficients at the corresponding divisors of Y, and hence $\kappa (\tilde {Y}) = -\infty $ . If Y has a non-RDP singularity, then $K_{\tilde {Y}}$ has negative coefficients at the corresponding exceptional curves, and hence $\kappa (\tilde {Y}) = -\infty $ . In either case, $\tilde {Y}$ is either ruled or rational. Suppose neither is the case. Then $K_Y$ is an RDP surface with numerically trivial $K_Y$ . Since we have

$$\begin{align*}H^0(Y^{\mathrm{sm}}, \Omega_Y^2) = H^0(\pi^{-1}(Y^{\mathrm{sm}}), \Omega_X^2)^G = H^0(X, \Omega_X^2)^G = 0, \end{align*}$$

$K_Y$ is not trivial. Then, by the classification of such surfaces (see Table 1), $\tilde {Y}$ is either an Enriques surface or a (quasi-)hyperelliptic surface. This settles the non-symplectic case.

Now, suppose G acts symplectically. By Proposition 3.4, Y is an RDP surface with $K_Y$ trivial. By the classification of surfaces with trivial canonical divisor, Y is an RDP K3 surface, an abelian surface, a non-classical RDP Enriques surface ( $p = 2$ ), or a (quasi-)hyperelliptic surface ( $p = 2,3$ ). (Note that abelian and (quasi-)hyperelliptic surface admit no smooth rational curves.)

Since Y has only RDP singularities, we have $h^1(\tilde {Y}, \mathcal {O}_{\tilde {Y}}) = h^1(Y, \mathcal {O}_Y)$ . Since $p \nmid \lvert G \rvert $ , $(\mathcal {O}_X)^G \subset \mathcal {O}_X$ is a direct summand and hence we have $h^1(Y, \mathcal {O}_Y) = h^1(Y, (\pi _* \mathcal {O}_X)^G) \leq h^1(Y, \pi _* \mathcal {O}_X) = h^1(X, \mathcal {O}_X)$ . Therefore, if X is a K3 surface, then Y cannot be a nonclassical RDP Enriques surface.

It remains to show that if X is an abelian surface, then Y cannot be a (quasi-)hyperelliptic surface. (If $p \neq 2,3$ , then this is clear since (quasi-)hyperelliptic surfaces always have nontrivial canonical divisor.) If X is an abelian surface and Y is a (quasi-)hyperelliptic surface, then since a (quasi-)hyperelliptic surface admits no smooth rational curves, no element of $G \setminus \{1\}$ has fixed points. It suffices to show that any fixed-point-free symplectic automorphism g of an abelian surface X of finite order not divisible by p is a translation, since the quotient of an abelian variety by a finite group of translations is an abelian variety.

Suppose g is such an automorphism. Endow X with a group variety structure (i.e., choose an origin) and write $g(x) = h(x) + a$ with h an automorphism of the group variety (i.e., h fixes the origin) and a a point. Then h is symplectic (since g and the translation by a are symplectic) and of finite order dividing $\operatorname {\mathrm {ord}}(g)$ , since $x = g^{\operatorname {\mathrm {ord}}(g)}(x) = h^{\operatorname {\mathrm {ord}}(g)}(x) + (h^{\operatorname {\mathrm {ord}}(g)-1}(a) + \dots + a)$ . If $h = \mathrm {id}$ , then g is a translation. Suppose $h \neq \mathrm {id}$ . Then h acts on the tangent space of each fixed point via $\operatorname {\mathrm {SL}}_2(k)$ (since h is symplectic and of finite order not divisible by p), and hence $\operatorname {\mathrm {Fix}}(h)$ is isolated. Hence, $h - \mathrm {id}$ has finite kernel and hence is surjective. Let x be a point with $h(x) - x = - a$ . Then $g(x) = x$ , contradiction.

6 $\mu _p$ -quotients of RDP K3 surfaces and abelian surfaces

The following theorems are the $\mu _p$ -analogue of Theorem 5.1.

Theorem 6.1. The quotient of an RDP K3 surface by a symplectic $\mu _{p}$ -action is again an RDP K3 surface.

Theorem 6.2. The quotient $X/\mu _p$ of an RDP K3 surface by a non-symplectic action of $\mu _p$ is either a rational surface (possibly with non-RDP singularities) or an RDP Enriques surface. The quotient is an RDP Enriques surface if and only if the action is fixed-point-free, and this can happen only if $p = 2$ .

Theorem 6.3. A $\mu _{p^e}$ -action on an abelian variety is always symplectic, in the sense that the one-dimensional space of top differential forms is of weight $0$ , and the quotient is again an abelian surface.

Proof of Theorem 6.3

It suffices to consider $\mu _p$ -actions. A $\mu _p$ -action on A corresponds [GP, Th. VII.7.2(ii)] to an element $v \in H^0(A, T) \cong T_0 A$ satisfying $v^p = v$ , and hence the action can be identified with the translation action by a subgroup scheme of A. Then the quotient is again an abelian variety, and hence this action is symplectic by Proposition 2.10.

Proof of Theorem 6.1

By Theorem 4.6(1,2), $Y = X/\mu _p$ is an RDP surface. Then, by Proposition 2.10, $K_{Y}$ is trivial, and by the classification of surfaces, Y is either an RDP K3 surface, an abelian surface, a nonclassical RDP Enriques surface, or a (quasi-)hyperelliptic surface.

Since ${\pi }:{X}\to {Y}$ is purely inseparable, we have $\dim H_{\mathrm {\acute et}}^i(X, \mathbb Q_l) = \dim H_{\mathrm {\acute et}}^i(Y, \mathbb Q_l)$ , in particular $b_1(Y) = b_1(X) = 0$ (where $b_i = \dim H_{\mathrm {\acute et}}^i$ is the ith Betti number). Since Y is an RDP surface and since $\mathcal {O}_Y = (\pi _* \mathcal {O}_X)_0$ is a direct summand of $\pi _* \mathcal {O}_X$ , we have $h^1(\tilde {Y}, \mathcal {O}_{\tilde {Y}}) = h^1(Y, \mathcal {O}_Y) \leq h^1(Y, \pi _* \mathcal {O}_X) = h^1(X, \mathcal {O}_X) = 0$ . Hence, Y is an RDP K3 surface.

Proof of Theorem 6.2

Let X be an RDP K3 surface equipped with a nontrivial non-symplectic $\mu _p$ -action. Let $Y = X/\mu _p$ . We have $b_1(Y) = b_1(X) = 0$ , and hence as in the tame case (§5), the minimal resolution $\tilde {Y}$ of Y cannot be abelian, (quasi-)hyperelliptic, nor non-rational ruled.

If D has non-isolated fixed points, then by the Rudakov–Shafarevich formula $K_X \sim (p-1) (D) + \pi ^* K_Y$ [RS, Cor. 1 to Prop. 3], or by Proposition 2.10, $\kappa (\tilde {Y}) = -\infty $ . If D has an isolated fixed point $w \in X$ , then by Theorem 4.6(3), $\pi (w) \in Y$ is a non-RDP singularity, and then hence $\kappa (\tilde {Y}) = -\infty $ . In either case, Y is a rational surface.

Now, assume D is fixed-point-free. Then, by Theorem 4.6(1), Y is an RDP surface and, by the Rudakov–Shafarevich formula, $K_{Y}$ is torsion. Moreover, it follows from Proposition 2.10 that the space $H^0(Y^{\mathrm {sm}}, (\Omega ^2)^{\otimes n})$ is $0$ if $0 < n < p$ and is generated by a nonvanishing multicanonical form if $n = p$ . Thus, $K_{\tilde {Y}}$ is nonzero and p-torsion. By the classification of surfaces, it follows that Y is an RDP Enriques surface. Then, since $2K_{Y} \sim 0$ and $H^0(Y^{\mathrm {sm}}, \mathcal {O}(n K_{Y})) = 0$ for $0 < n < p$ , we have $p = 2$ .

There is also the following relation with the height of K3 surfaces. The height is an invariant of a K3 surface in positive characteristic which is either $\infty $ or an integer in $\{1, \dots , 10\}$ . See §8 for more details.

Corollary 6.4. Let X be an RDP K3 surface in characteristic p equipped with a nontrivial $\mu _p$ -action. If X is of finite height, then the action is symplectic and the quotient is an RDP K3 surface.

Proof. We assume that the action is non-symplectic and show that then X is not of finite height. By Theorem 6.2, the quotient $Y = X / \mu _p$ is a rational surface or an RDP Enriques surface. Hence, X admits a purely inseparable finite morphism ${Y}^{(1/p)} \to X$ from a rational surface or an RDP Enriques surface. Hence, $H_{\mathrm {\acute et}}^2(X, \mathbb Q_l)$ , which is isomorphic to $H^2({Y}^{(1/p)}, \mathbb Q_l)$ , is generated by algebraic cycles, which is impossible if X is of finite height by Lemma 8.3(2).

Remark 6.5. In a subsequent paper [Mat4, Th. 1.3], we prove that also the converse holds, and we moreover determine the height in terms of the singularities of X and Y.

We call a proper birational morphism $X' \to X$ between RDP surfaces to be a partial resolution if it is dominated by the minimal resolution $\tilde {X}$ of X. The following proposition often enables us to reduce assertions on $\mu _p$ -actions to simpler cases.

Proposition 6.6. Let X be an RDP surface equipped with a $\mu _p$ -action.

1. 1. Among partial resolutions of X to which the $\mu _p$ -action extends, there exists a unique maximal one, which we call the maximal partial resolution of X.

2. 2. A partial resolution $X' \to X$ is maximal if and only if it satisfies the property $\operatorname {\mathrm {Sing}}(X') \cap \pi ^{-1}(\operatorname {\mathrm {Sing}}(X'/\mu _p)) = \emptyset $ .

3. 3. The action on X is fixed-point-free if and only if the action on the maximal partial resolution is fixed-point-free.

4. 4. Suppose X is an RDP K3 surface. The action on X is symplectic if and only if the action on the maximal partial resolution is symplectic.

We say that an RDP surface X equipped with a $\mu _p$ -action is maximal if it is the maximal partial resolution of itself.

Proof. Note that isomorphism classes of partial resolutions are in one-to-one correspondence to subsets of the set of exceptional curves of $\tilde {X} \to X$ .

(1) Let $X'$ be a partial resolution. Let $\tilde {D}$ and $D'$ be the rational derivations induced by D on $\tilde {X}$ and $X'$ . Clearly, $D'$ is regular and thus corresponds to a $\mu _p$ -action if and only if the coefficient of $(D')$ for each exceptional curve of $X' \to X$ is nonnegative. Therefore, the contraction of all exceptional curves on $\tilde {X}$ with negative coefficients in $(\tilde {D})$ is the maximal partial resolution.

(2) First, suppose $X'$ does not satisfy the property. There is an RDP $w \in X'$ such that $\pi (w)$ is not smooth. If w is a fixed RDP, then let $X^{\prime }_1 = \operatorname {\mathrm {Bl}}_w X'$ . If w is a non-fixed RDP, then let $X^{\prime }_1 = X' \times _{Y'} Y^{\prime }_1$ , where $Y'$ is the $\mu _p$ -quotient of $X'$ and $Y^{\prime }_1 \to Y'$ is the minimal resolution at $\pi (w)$ . Then the $\mu _p$ -action extends to $X^{\prime }_1$ (by Proposition 2.8 in the former case, clear in the latter case) and $X^{\prime }_1$ is a partial resolution of $X'$ (clear in the former case, by Theorem 4.6(1) in the latter case). Thus, $X'$ is not maximal.

Conversely, suppose $X'$ satisfies the property. Let $w \in \operatorname {\mathrm {Sing}}(X')$ . To show that $X'$ is maximal, it suffices to show that every exceptional curve of $\tilde {X}$ above w appears in $(\tilde {D})$ with negative coefficient, since then it must be contracted in the maximal partial resolution. This follows from the Rudakov–Shafarevich formula $K_{\tilde {X}} \sim (p-1) (\tilde {D}) + \pi ^* K_{\tilde {Y}}$ [RS, Cor. 1 to Prop. 3], where $\tilde {Y} = \tilde {X}/\mu _p$ : for each exceptional curve, its coefficient in $K_{\tilde {X}}$ (resp. $\pi ^* K_{\tilde {Y}}$ ) is $0$ (resp. positive) since $\tilde {X}$ is the minimal resolution of the RDP w (resp. since $\pi (w)$ is a smooth point). Alternatively, one can use the explicit computation of $(\tilde {D})$ given in Lemma 6.7. (w is non-fixed since $\pi (w)$ is a smooth point [Theorem 4.6(2,3)].)

We also observe that the maximal partial resolution of X can be constructed by repeatedly applying the procedure of constructing $X_1'$ from $X'$ . Indeed, the total index of RDPs strictly decreases through the procedure.

(3) It suffices to show that the procedure above preserves the (non-)fixed-point-freeness. If w is a non-fixed RDP, then there is no fixed point above w by the functoriality of the fixed-point scheme. If w is a fixed RDP, then there is a fixed point above w by Corollary 2.12.

(4) The spaces of $2$ -forms in question are naturally isomorphic.

Lemma 6.7. Let X be an RDP surface, and let D be a derivation of multiplicative type. Let $w \in X$ be a non-fixed RDP of index n, and suppose the image of w is a smooth point. Let $\tilde X$ be the minimal resolution at w, and let $\tilde D$ be the induced rational derivation on $\tilde X$ . Let $(\tilde {D})$ and $\langle \tilde {D} \rangle $ be the divisorial and isolated fixed locus of $\tilde D$ above w. Then every exceptional curve of $\tilde {X}$ above w appears in $(\tilde {D})$ with negative coefficient, and we have $\deg \langle \tilde {D} \rangle = \frac {p-2}{p-1} n$ and $(\tilde {D})^2 = - \frac {2}{p-1} n$ .

Proof. We compute $(\tilde {D})$ and $\langle \tilde {D} \rangle $ by a straightforward calculation using the classification of non-fixed RDPs given in Proposition 4.7. See [Mat3, Lem. 3.11] for the result. Then the assertions follow.

Proposition 6.8. Suppose each of X and Y is either an RDP K3 surface or an RDP Enriques surface. Let $\pi \colon X \to Y$ be a $\mu _p$ -quotient morphism. Suppose X is maximal with respect to the $\mu _p$ -action. Then the total index $N_1$ of non-fixed RDPs and the number $N_2$ of fixed points on X are as follows:

$$\begin{align*}(N_1, N_2) = \begin{cases} (24 \frac{p-1}{p+1}, 24 \frac{1}{p+1}), & \text{if } (\tilde X, \tilde Y) \text{ is } (\mathrm{K3},\mathrm{K3}), \\ (12 \frac{2p-1}{p+1}, 12 \frac{p-2}{p^2-1}) = (12, 0), & \text{if } (\tilde X, \tilde Y) \text{ is } (\mathrm{K3},\mathrm{Enr}), \\ (12 \frac{p-2}{p+1}, 12 \frac{2p-1}{p^2-1}) = (0, 12), & \text{if } (\tilde X, \tilde Y) \text{ is } (\mathrm{Enr},\mathrm{K3}), \\ (12 \frac{p-1}{p+1}, 12 \frac{1}{p+1}), & \text{if } (\tilde X, \tilde Y) \text{ is } (\mathrm{Enr},\mathrm{Enr}). \\ \end{cases} \end{align*}$$

In the cases where $(\tilde {X}, \tilde {Y})$ is $(\mathrm {K3}, \mathrm {Enr})$ or $(\mathrm {Enr}, \mathrm {K3})$ , only $p = 2$ is possible.

Proof. Let D be the corresponding derivation of multiplicative type. Since $\tilde {Y}$ is of Kodaira dimension $0$ , $\operatorname {\mathrm {Fix}}(D)$ consists only of isolated points (possibly none).

Define the rational derivation $\tilde D$ on $\tilde {X}$ as in Lemma 6.7. Since the fixed locus of $\tilde D$ consists of those above non-fixed RDPs on X and the $0$ -cycle of fixed points on X, by Lemma 6.7, we have

$$\begin{align*}(\tilde{D})^2 = - \frac{2}{p-1} N_1, \quad \deg \langle \tilde{D} \rangle = N_2 + \frac{p-2}{p-1} N_1, \end{align*}$$

and by the Katsura–Takeda formula (Proposition 2.13), we have

$$\begin{align*}\deg c_2(\tilde{X}) = (N_2 + \frac{p-2}{p-1} N_1) + 0 + \frac{2}{p-1} N_1 = N_2 + \frac{p}{p-1} N_1. \end{align*}$$

On the other hand, we have $\dim H_{\mathrm {\acute et}}^2(X, \mathbb Q_l) = \dim H_{\mathrm {\acute et}}^2(\tilde {X}, \mathbb Q_l) - N_1 = b_2(\tilde {X}) - N_1$ . Since Y is an RDP surface whose RDPs are precisely the images (which are $A_{p-1}$ ) of the fixed smooth points of X, we have $\dim H_{\mathrm {\acute et}}^2(Y, \mathbb Q_l) = \dim H_{\mathrm {\acute et}}^2(\tilde {Y}, \mathbb Q_l) - (p-1)N_2 = b_2(\tilde {Y}) - (p-1)N_2$ . Since $X \to Y$ is purely inseparable, we have $\dim H_{\mathrm {\acute et}}^2(X, \mathbb Q_l) = \dim H_{\mathrm {\acute et}}^2(Y, \mathbb Q_l)$ . Therefore, we have $(p-1) N_2 - N_1 = b_2(\tilde {Y}) - b_2(\tilde {X})$ .

Combining the two equalities, we determine $(N_1,N_2)$ in terms of p. In two cases, only $p = 2$ is possible since $12 \frac {p-2}{p^2-1}$ or $12 \frac {2p-1}{p^2-1}$ must be an integer.

7 Possible orders of symplectic $\mu _n$ -actions on RDP K3 surfaces

The following theorem is again parallel to the case of automorphisms of finite tame order, but the proof (for $\mu _p$ -actions) is quite different.

Theorem 7.1. Let X be an RDP K3 surface in characteristic p equipped with a symplectic $\mu _{n}$ -action ( $n> 1$ , divisible by p or not). Then:

1. 1. $n \leq 8$ .

2. 2. The number of fixed points, counted with the multiplicities defined after Proposition 4.14, depends only on n and is as in Table 7.

   Table 7 Symplectic $\mu _n$ -actions on K3 surfaces.

   

3. 3. Assume all fixed points are smooth points. Then the decomposition of $\bigoplus _{w \in \operatorname {\mathrm {Fix}}(\mu _n)} T^*_w X$ with respect to the $\mu _n$ -action is concentrated on the subset $(\mathbb Z/n\mathbb Z)^* \subset \mathbb Z/n\mathbb Z$ , and for every $i \in (\mathbb Z/n\mathbb Z)^*$ , the ith summand has dimension as in Table 7.

If $p \nmid n$ (resp. if $n = p$ ), then assertion (3) means that every primitive nth root of $1$ (resp. every element of $\mathbb F_p^*$ ) appears as an eigenvalue of a fixed generator of $\mu _n \cong \mathbb Z/n\mathbb Z$ (resp. of the corresponding derivation) on the space $\bigoplus _{w \in \operatorname {\mathrm {Fix}}(\mu _n)} T^*_w X$ with equal multiplicity.

For each p and each $n \leq 8$ , there indeed exists an RDP K3 surface equipped with a symplectic $\mu _{n}$ -action in characteristic p. See §9.1 for explicit examples.

Proof of Theorem 7.1 for the case $p \nmid n$

We may assume X is smooth.

Assertions (1) and (2) are proved by Nikulin [Ni, §5] ( $p = 0$ ) and Dolgachev–Keum [DK2, Th. 3.3] ( $p> 0$ ). (Both proofs overlooked the case $n = 14$ , but their arguments for the nonexistence of the case $n = 15$ apply to case $n = 14$ .)

(3) (If $p = 0$ , then this follows from the argument in [Mu, Prop. 1.2]. We give another proof, applicable to all $p \geq 0$ .)

Let $w \in X$ , and let $\mu _r = \operatorname {\mathrm {Stab}}(w) \subset \mu _n$ be its stabilizer group. Suppose the decomposition of $T^*_w X$ ( $ = \mathfrak {m}_w / \mathfrak {m}_w^2$ ) with respect to the $\mu _r$ -action is concentrated on two (not necessarily distinct) weights $j_1, j_2 \in \mathbb Z/r\mathbb Z$ . Then since the action on $\Omega ^2_{X,w} \cong \bigwedge ^2 T^*_w X$ is trivial, we have $j_1 + j_2 = 0$ . We have $j_1,j_2 \in (\mathbb Z/r\mathbb Z)^*$ , since if $\gcd (j_1, r) = \gcd (j_2, r) = r'> 1$ , then $\mu _{r'}$ acts trivially, contradicting the assumption that the action is faithful. This already proves the assertion if $n = 2,3,4,6$ , since up to sign there is only one element in $(\mathbb Z/n\mathbb Z)^*$ .

For each divisor $r \neq 1$ of n and for each $0 \leq j \leq \lfloor r/2 \rfloor $ , let $S_{r,j}$ be the set of the points $w \in X$ with $\operatorname {\mathrm {Stab}}(w) = \mu _r$ and with $\mu _r$ acting on $T^*_w X$ by weights j and $-j$ . Then $S_{r,j}$ is a finite set and it is empty if $j \not \in (\mathbb Z/r\mathbb Z)^*$ . Let $\tilde {S}_{r,j} = S_{r,j} / \mu _n$ be the set of $\mu _n$ -orbits of points of $S_{r,j}$ . Let $N_{r,j} = \lvert S_{r,j} \rvert $ and $\tilde {N}_{r,j} = \lvert \tilde {S}_{r,j} \rvert = N_{r,j} / (n/r)$ .

Let ${\rho }:{\tilde {Y}}\to {Y = X / \mu _n}$ be the minimal resolution (then $\tilde {Y}$ is a smooth K3 surface), and let $\pi ' \colon X' = X \times _Y \tilde {Y} \to \tilde {Y}$ . Let $\pi ^{\prime }_* \mathcal {O}_{X'} = \bigoplus _{i \in \mathbb Z/n\mathbb Z} (\pi ^{\prime }_* \mathcal {O}_{X'})_i$ be the decomposition induced by the $\mu _n$ -action. Then, by Lemma 4.15, $(\pi ^{\prime }_* \mathcal {O}_{X'})_i$ are invertible sheaves, and they are described as follows. For each $i \in \mathbb Z/n\mathbb Z$ , let $C_i$ be the corresponding class of Cartier divisors. For each orbit $w \in \tilde {S}_{r,j}$ , its image $\pi (w)$ is an RDP of type $A_{r-1}$ , and let $e_{w,k}$ ( $k = 1, \dots , r-1$ ) be the exceptional curves in $\tilde {Y}$ above $\pi (w)$ , ordered in a way that $e_{w,k} \cap e_{w,k'} \neq \emptyset $ if and only if $\lvert k - k' \rvert \leq 1$ . Then, after possibly reversing the ordering, we have a linear equivalence

$$\begin{align*}- r C_i \sim \sum_{r \mid n, r \neq 1} \sum_{j = 0}^{\lfloor r/2 \rfloor} \sum_{w \in \tilde{S}_{r,j}} \sum_{k = 1}^{r-1} {f_r(({j^{-1}} i \bmod r), k)} e_{w,k} \end{align*}$$

(see Lemma 4.15 for the definition of the function ${f_r}:{\{1, \dots , r-1\}^2}\to {\mathbb Z}$ ) for each $i \neq 0$ .

Let m be any integer with $1 \leq m \leq r-1$ . Using the equality

$$\begin{align*}\Biggl( \sum_{k=1}^{r-1} \frac{f_r(m,k)}{r} e_{w,k} \Biggr) \cdot e_{w,k'} = \begin{cases} -1, & \text{if } k' = r - m, \\ 0, & \text{otherwise,} \end{cases} \end{align*}$$

we compute that

$$\begin{align*}\Biggl( \sum_{k=1}^{r-1} \frac{f_r(m,k)}{r} e_{w,k} \Biggr)^2 = -\frac{m(r-m)}{r}. \end{align*}$$

Hence, we have

$$\begin{align*}- C_i^2 = \sum_{r \mid n, r \neq 1} \sum_{j = 0}^{\lfloor r/2 \rfloor} \tilde{N}_{r,j} \cdot \frac{({j^{-1}}i \bmod r)(r - {j^{-1}}i \bmod r)}{r} \end{align*}$$

and this must belong to $2 \mathbb Z$ .

Assume $n = 5$ . Then we have $N_{5,j} = \tilde {N}_{5,j}$ , $N_{5,1} + N_{5,2} = 4$ , and

$$\begin{align*}-C_1^2 = \frac{4}{5} N_{5,1} + \frac{6}{5} N_{5,2} \in 2 \mathbb Z. \end{align*}$$

Hence, $(N_{5,1}, N_{5,2}) = (2, 2)$ .

Assume $n = 7$ . Then we have $N_{7,j} = \tilde {N}_{7,j}$ , $N_{7,1} + N_{7,2} + N_{7,3} = 3$ , and

$$\begin{align*}-C_1^2 = \frac{6}{7} N_{7,1} + \frac{12}{7} N_{7,2} + \frac{10}{7} N_{7,3} \in 2 \mathbb Z. \end{align*}$$

Hence, $(N_{7,1}, N_{7,2}, N_{7,3}) = (1, 1, 1)$ .

Assume $n = 8$ . By assertion (2) for the cases $n = 2,4,8$ , we have $\tilde {N}_{2,1} = 1$ , $\tilde {N}_{4,1} = 1$ , $\tilde {N}_{8,1} + \tilde {N}_{8,3} = 2$ , and

$$\begin{align*}-C_1^2 = \frac{1}{2} \tilde{N}_{2,1} + \frac{3}{4} \tilde{N}_{4,1} + \frac{7}{8} \tilde{N}_{8,1} + \frac{15}{8} \tilde{N}_{8,3} \in 2 \mathbb Z. \end{align*}$$

Hence, $({N}_{8,1}, {N}_{8,3}) = (\tilde {N}_{8,1}, \tilde {N}_{8,3}) = (1, 1)$ .

Remark 7.2. By above, we have $C_i^2 = -4$ for any $2 \leq n \leq 8$ and any $1 \leq i \leq n-1$ . As we see below, this holds also if p divides n. This implies $\chi (\tilde {Y}, (\pi ^{\prime }_* \mathcal {O}_{X'})_i) = 0$ for $i \neq 0$ . Then we obtain $\chi (Y, (\pi _* \mathcal {O}_{X})_i) = 0$ for $i \neq 0$ , since $\rho _*((\pi ^{\prime }_* \mathcal {O}_{X'})_i) = (\pi _* \mathcal {O}_{X'})_i$ and $R^q \rho _*((\pi ^{\prime }_* \mathcal {O}_{X'})_i) = 0$ for $q> 0$ . This is finer than the equality

$$\begin{align*}\sum_{i \neq 0} \chi(Y, (\pi_* \mathcal{O}_{X})_i) = \chi(X, \mathcal{O}_X) - \chi(Y, \mathcal{O}_Y) = 2 - 2 = 0. \end{align*}$$

Proof of Theorem 7.1 for the case $n = p$

We may assume that X is maximal. (For assertion (2), the multiplicity is by definition compatible with blowups at fixed points.) This means that all fixed points are smooth points, and that the singularities of the quotient surface Y are all $A_{p-1}$ and are precisely the images of the fixed points. Let D be the corresponding derivation.

As in the previous case, let $\tilde {Y}$ be the minimal resolution of Y (hence a smooth K3 surface) and let $\pi ' \colon X' = X \times _Y \tilde {Y} \to \tilde {Y}$ . The sheaf $\pi ^{\prime }_* \mathcal {O}_{X'}$ admits a decomposition to invertible sheaves $(\pi ^{\prime }_* \mathcal {O}_{X'})_i$ . For each $i \in \mathbb Z/p\mathbb Z$ , let $C_i$ be the corresponding class of Cartier divisor. As in the previous case, we have

$$\begin{align*}- C_i^2 = \sum_{j = 0}^{\lfloor p/2 \rfloor} N_{p,j} \cdot \frac{({j^{-1}}i \bmod p)(p - {j^{-1}}i \bmod p)}{p}. \end{align*}$$

By Proposition 6.8, we have $N := \sum _{j} N_{p,j} = 24/(p+1)$ . Hence, $p \in \{2,3,5,7,11,23\}$ . If $p = 23,$ then $N = 1$ and the exceptional curves generate a negative definite sublattice of rank $p-1 = 22$ of the indefinite lattice $\operatorname {\mathrm {Pic}}(\tilde {Y})$ of rank $\leq 22$ , contradiction. If $p = 11$ , then $N = 2$ and then $C_i^2$ (for any $i \in (\mathbb Z/p\mathbb Z)^*$ ) cannot be an integer since the sum of two nonzero squares in $\mathbb F_{11}$ cannot be zero. Hence, we have $p \in \{2,3,5,7\}$ , and we can determine the multiplicities of the weights as in the $p \nmid n$ case.

Corollary 7.3. Let X be an RDP K3 surface equipped with a symplectic $\mu _{q}$ -action with $q = 5,7$ . Here, both $p = q$ and $p \neq q$ are allowed.

• • If $q = 7$ , then any fixed point is a smooth point.

• • If $q = 5$ , then any fixed point is a smooth point or an RDP of type $A_1$ .

Proof. Let $w \in X$ be a fixed point. By Proposition 4.14, w is of type $A_{m-1}$ for some $m \geq 1$ . Let $\pm i \in (\mathbb Z/q\mathbb Z)^*$ be the nonzero weights of $\mathfrak {m}_w/\mathfrak {m}_w^2$ with respect to the $\mu _q$ -action. Let $\tilde {X}$ be the minimal resolution of X at w (to which the $\mu _q$ -action extends). One can calculate the local equation to show that all (smooth) fixed point of $\tilde {X}$ above w has weights $\pm i$ . Since there are m such points, it follows from Theorem 7.1(3) that $m \leq 1$ if $q = 7$ and $m \leq 2$ if $q = 5$ .

Proof of Theorem 7.1 for the case $n = p^e$ ( $e \geq 2$ )

For each $0 \leq j \leq e$ , let $\pi _j \colon X \to X_j = X/\mu _{p^j}$ be the quotient morphism by the subgroup scheme $\mu _{p^j} \subset \mu _{p^e}$ , and for each $0 \leq j \leq e-1$ , let $D_j$ be the derivation on $X_j$ corresponding to the action of $\mu _{p^{j+1}}/\mu _{p^j}$ .

Let $w \in X$ be a $\mu _p$ -fixed point. Let $\mu _{p^f} = \operatorname {\mathrm {Stab}}(w)$ (then $1 \leq f \leq e$ ). Then, by Remark 4.4, $\mu _{p^f}$ acts symplectically at w. By Proposition 4.13, either w is of type $A_{m-1}$ for some $m \geq 1$ with $p^{e-f} \mid m$ , or w is $D_{2m+1}^{m-1}$ or $E_7^2$ or $E_8^3$ and $p^e = 4$ . (Again, we use the convention that a smooth point is of type $A_0$ .) Then since each $D_j$ ( $j < f$ ) is symplectic at $\pi _j(w)$ , we observe that $\pi _f(w) \in X_f$ is of type $A_{p^e m - 1}$ or $D_5^0$ or $D_{2m+1}^{m-1}$ . (Since X has a $\mu _p$ -fixed point and since $X_f$ is an RDP K3 surface, this already implies $p^e - 1 < 22$ .)

By Lemma 4.11, any preimage of any fixed point of $D_{j}$ is again fixed. In other words, the fixed points of $D_j$ on $X_j$ are precisely the images of the $\mu _{p^{j+1}}$ -fixed points on X.

For each $1 \leq f \leq e$ , let $\tilde {S}_f \subset X$ be the points with stabilizer equal to $\mu _{p^f}$ . For each $w \in \tilde {S}_f$ , let $m(w)$ be its multiplicity of w defined after Proposition 4.14 with respect to the $\mu _{p^f}$ -action. Then, by Propositions 4.13 and 4.14, we have $p^{e-f} \mid m(w)$ . Let $M_f = \sum _{w \in \tilde {S}_f} m(w)$ for each $1 \leq f \leq e$ , then by above $p^{e-f} \mid M_f$ . Using the equality mentioned after Proposition 4.14 and assertion (2) for $D_{e-1}$ and $D_{e-2}$ on $X_{e-1}$ and $X_{e-2}$ , we obtain $p^{e-1} M_e = p^{e-2} (M_e + M_{e-1}) = 24/(p+1)$ , and hence $M_e = 24 / (p^{e-1}(p+1))$ and $M_{e-1} / p = 24 (p-1) / (p^{e}(p+1))$ . Since $M_{e-1} / p$ is an integer, $p^e$ divides $24$ . Therefore, $p^e = 2^2, 2^3$ . Moreover, we obtain $M_f = p^{e-f} \cdot 24 (p-1) / (p^{e}(p+1))$ ( $1 \leq f \leq e-1$ ) by applying assertion (2) to $D_{j}$ ( $0 \leq j \leq e-1$ ).

Assertion (3) is trivial if $n = 4$ . Suppose $n = 8$ . For each $1 \leq f \leq 3$ and $0 \leq j \leq 2^f/2$ , let $\tilde {S}_{2^f,j}$ be the set of the points with stabilizer $\mu _{2^f}$ and with primitive weights $\pm j \in (\mathbb Z/2^f\mathbb Z)^*$ , and let $\tilde {N}_{2^f,j} = (2^{e-f})^{-1} \sum _{w \in \tilde {S}_{2^f,j}} m(w)$ . We have $\sum _{j} \tilde {N}_{2^f,j} = (2^{e-f})^{-1} M_f$ for each $1 \leq~f~\leq~e$ . Then we again have $\tilde {N}_{2,1} = 1$ , $\tilde {N}_{4,1} = 1$ , $\tilde {N}_{8,1} + \tilde {N}_{8,3} = 2$ , and

$$\begin{align*}-C_1^2 = \frac{1}{2} \tilde{N}_{2,1} + \frac{3}{4} \tilde{N}_{4,1} + \frac{7}{8} \tilde{N}_{8,1} + \frac{15}{8} \tilde{N}_{8,3} \in 2 \mathbb Z. \end{align*}$$

Hence, $(\tilde {N}_{8,1}, \tilde {N}_{8,3}) = (1,1)$ .

Proof of Theorem 7.1 for the remaining cases

First, we show that if $n = pq$ where q is a prime $\neq p$ , then $n = 6$ . We have $\mu _n = \mu _p \times \mu _q \cong \mu _p \times \mathbb Z/q\mathbb Z$ . We may assume that X is maximal with respect to the $\mu _p$ -action. Let $\pi _q \colon X \to X_q = X / \mu _q$ and $\pi _p \colon X \to X_p = X / \mu _p$ . Note that $w \in X$ is fixed by the $\mu _p$ -action if and only if $\pi _q(w) \in X_q$ is fixed by the $\mu _p$ -action. Let $a_1$ and $a_q$ be the number of $\mu _q$ -orbits of length $1$ and q of $\mu _p$ -fixed points of X (which are all smooth by assumption). Then the $\mu _p$ -fixed points of $X_q$ consists of $a_1$ points of type $A_{q-1}$ and $a_q$ smooth points. Applying assertion (2) to the $\mu _p$ -actions on X and $X_q$ we have $a_1 + q a_q = q a_1 + a_q = 24/(p+1)$ . Therefore, $a_1 = a_q = 24/(p+1)(q+1)$ and hence $(a_1, \{p, q \}) = (2, \{2, 3 \}), (1, \{2, 7 \} ), (1, \{3, 5 \})$ .

The cases $(a_1, \{p, q \}) = (1, \{2, 7 \} ), (1, \{3, 5 \})$ are impossible since letting $w \in X$ be the unique $\mu _{pq}$ -fixed point (which is a smooth point), if $pq = 14$ then $\pi _2(w) \in X_2$ is a $\mu _7$ -fixed RDP of type $A_{1}$ , and if $pq = 15$ then $\pi _3(w) \in X_3$ is a $\mu _5$ -fixed RDP of type $A_{2}$ , both contradicting Corollary 7.3.

Now, we consider general n. It remains to show that the cases $(p,n) = (2,12), (3,12)$ are impossible.

Assume $(p,n) = (3,12)$ . As above, we may assume X is maximal with respect to the $\mu _3$ -action. There are exactly six $\mu _3$ -fixed points, all smooth. By the above argument for $(p,n) = (3,6)$ , exactly two of them are $\mu _2$ -fixed, and among the images of these two points in $X/\mu _2$ , exactly one is $(\mu _4/\mu _2)$ -fixed. This is impossible since non- $(\mu _4/\mu _2)$ -fixed points in $X / \mu _2$ come by pairs.

Now, assume $(p,n) = (2,12)$ . As in the proof of the $n = p^e$ case (applied to the $\mu _4$ -action), let $\tilde {S}_1$ be the set of $\mu _2$ -fixed non- $\mu _4$ -fixed points, and then we have $M_1 = \sum _{w \in \tilde {S}_1} m(w) = 4$ and $2 \mid m(w)$ . Hence, $\lvert \tilde {S}_1 \rvert $ is $1$ or $2$ . Since the $\mu _3$ -action on X preserves this $1$ - or $2$ -point set $\tilde {S}_1$ , it acts on $\tilde {S}_1$ trivially, and hence fixes at least four $\mu _2$ -fixed points (counted with multiplicity $m(w)$ ), contradicting the observation $a_1 = 2$ for $\mu _6$ -actions.

Assertion (3) for $n = 6$ is trivial.

8 Possible orders of $\mu _n$ -actions on RDP K3 surfaces

Let $S_{\mathrm {cyc}}(p)$ (resp. $S_{\mu }(p)$ ) be the set of positive integers n for which there exists an RDP K3 surface equipped with an automorphism of order n (resp. a $\mu _n$ -action) in characteristic p. We clearly have $S_{\mathrm {cyc}}(0) = S_{\mu }(0)$ and $S_{\mathrm {cyc}}(p)^{p'} = S_{\mu }(p)^{p'}$ , where $(-)^{p'}$ denotes the subset of prime-to-p elements.

Remark 8.1. Keum [Ke, Main Theorem] proved the following results on $S_{\mathrm {cyc}}(p)$ . (This set is denoted by $\mathrm {Ord}_p$ in his paper.) The sets $S_{\mathrm {cyc}}(p)$ for $p \neq 2,3$ are given by

$$ \begin{align*} &\qquad\qquad\qquad\qquad\qquad \ S_{\mathrm{cyc}}(0) = \{n : \phi(n) \leq 20\} \\ &= \{1, \dots, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 36, 38, 40, 42, 44, 48, 50, 54, 60, 66\}, \end{align*} $$

and

$$\begin{align*}S_{\mathrm{cyc}}(p) = \begin{cases} S_{\mathrm{cyc}}(0), & \text{if } p = 7 \text{ or } p \geq 23, \\ S_{\mathrm{cyc}}(0) \setminus \{p,2p\}, & \text{if } p = 13,17,19, \\ S_{\mathrm{cyc}}(0) \setminus \{44\}, & \text{if } p = 11, \\ S_{\mathrm{cyc}}(0) \setminus \{25,50,60\}, & \text{if } p = 5. \\ \end{cases} \end{align*}$$

Moreover, $S_{\mathrm {cyc}}(p)^{p'} = S_{\mathrm {cyc}}(0)^{p'}$ for all $p \geq 2$ . (The sets $S_{\mathrm {cyc}}(2)$ and $S_{\mathrm {cyc}}(3)$ are not determined.)

In this section, we determine the set $S_{\mu }(p)$ for all p.

Theorem 8.2. We have

$$\begin{align*}S_{\mu}(p) = \begin{cases} S_{\mathrm{cyc}}(0), & \text{if } p \neq 2,3,5,11, \\ S_{\mathrm{cyc}}(0) \setminus \{33,66\}, & \text{if } p = 11, \\ S_{\mathrm{cyc}}(0) \setminus \{25,40,50\}, & \text{if } p = 5, \\ S_{\mathrm{cyc}}(0) \setminus \{27,33,48,54,66\}, & \text{if } p = 3, \\ S_{\mathrm{cyc}}(0) \setminus \{34,40,44,48,50,54,66\}, & \text{if } p = 2. \end{cases} \end{align*}$$

In particular, there exists an RDP K3 surface equipped with a nontrivial $\mu _p$ -action in characteristic p if and only if $p \leq 19$ .

We need some preparations. The height h of a K3 surface X in characteristic $p> 0$ , whose definition we do not recall here, is either $\infty $ or an integer in $\{1, \dots , 10\}$ , and X is called supersingular or of finite height, respectively. If $h < \infty $ , then the inequality $\rho \leq 22 - 2h$ holds (Lemma 8.3(2)), where $\rho = \operatorname {\mathrm {rank}} \operatorname {\mathrm {Pic}}(X)$ is the Picard number. This implies that if $\rho \geq 21$ , then X is supersingular.

In fact, the Tate conjecture for K3 surfaces, now a theorem, states conversely that if X is supersingular, then $\rho = 22$ (see Lemma 8.3(4) for references). In this case, the $\mathbb Z_p$ -lattice $H_{\mathrm {crys}}^2(X/W(k))^{F = p}$ is isomorphic to $\operatorname {\mathrm {Pic}}(X) \otimes \mathbb Z_p$ [O2, Cor. 1.6], and the discriminant group of $H_{\mathrm {crys}}^2(X/W(k))^{F = p}$ (isomorphic to the discriminant group of $\operatorname {\mathrm {Pic}}(X)$ ) is of the form $(\mathbb Z/p\mathbb Z)^{2\sigma _0}$ for an integer $\sigma _0 \in \{1, \dots , 10\}$ . This $\sigma _0$ is called the Artin invariant of X. Here, the discriminant group of a nondegenerate lattice (resp. nondegenerate $\mathbb Z_p$ -lattice) L is defined to be the finite group $L^*/L$ , where $L^* = \operatorname {\mathrm {Hom}}_{\mathbb Z}(L, \mathbb Z)$ (resp. $L^* = \operatorname {\mathrm {Hom}}_{\mathbb Z_p}(L, \mathbb Z_p)$ ) is the dual of L.

We define the crystalline transcendental lattice $T(X) = T_{\mathrm {crys}}(X) \subset H_{\mathrm {crys}}^2(X/W(k))$ to be the orthogonal complement of the image of $\operatorname {\mathrm {Pic}}(X) \otimes W(k)$ , where $W(k)$ is the ring of Witt vectors over k. We have $\rho + \operatorname {\mathrm {rank}} T(X) = 22$ .

We collect some facts.

Lemma 8.3. Let X be a K3 surface in characteristic $p> 0$ .

1. 1. $\operatorname {\mathrm {Aut}}(X)$ acts on $H_{\mathrm {crys}}^2(X/W(k))$ and $H_{\mathrm {\acute et}}^2(X, \mathbb Z_l)$ (for any prime $l \neq p$ ) faithfully, and the characteristic polynomial of any element is independent of the cohomology (and l) and has coefficients in $\mathbb Q$ .

2. 2. If X is of finite height h, then $\rho \leq 22 - 2 h$ .

3. 3. Let $g \in \operatorname {\mathrm {Aut}}(X)$ and suppose it acts on $H^0(X, \Omega ^2)$ by a primitive Nth root of $1$ . If X is of finite height and $p \geq 3$ , then the characteristic polynomial of $g^*$ on $T_{\mathrm {crys}}(X)$ is the product of cyclotomic polynomials $\Phi _{N p^{e_i}}$ with nonnegative integers $e_i$ . In particular, $\phi (N) \mid \operatorname {\mathrm {rank}} T_{\mathrm {crys}}(X)$ , in particular $\operatorname {\mathrm {rank}} T_{\mathrm {crys}}(X) \geq \phi (N)$ .

4. 4. If X is supersingular, then $\rho = 22$ .

5. 5. Let $g \in \operatorname {\mathrm {Aut}}(X)$ and define N as in (3). If X is supersingular of Artin invariant $\sigma _0$ , then $N \mid (p^{\sigma _0} + 1)$ .

An immediate consequence of (5) is that, letting $g \in \operatorname {\mathrm {Aut}}(X)$ and N be as in (3), if there exists no integer $\sigma _0$ with $N \mid (p^{\sigma _0} + 1)$ , then X is not supersingular. This applies to, for example, $(p,N) = (3,8), (5,4)$ .

Proof. (1) [Ke, Th. 1.4].

(2) [I, Prop. II.5.12].

(3) See [Mat2, Lem. 2.4(3)], which deduces the assertion from [J, Th. 3.2].

(4) This assertion, the Tate conjecture for supersingular K3 surfaces, is proved by Madapusi Pera [Mad, Th. 1] for characteristic $\geq 3$ and by Kim and Madapusi Pera [KMP, Th. A.1] for characteristic $2$ .

The assertion under the assumption that X admits an elliptic fibration, which is true, for example, if $\rho \geq 5$ (which is always the case when we use this assertion in this paper), was proved much earlier by Artin [A1, Th. 1.7].

(5) This is proved by Nygaard [Ny2, Th. 2.1] under the assumption $p \neq 2$ . The argument is in fact valid for $p = 2$ (see [Mat2, Rem. 2.2]).

Proof of Theorem 8.2

Let $S_{\mu }'(p)$ be the set on the right-hand side of the statement. If n is a positive integer not divisible by p, then $\mu _n$ -action is equivalent to the action a cyclic group of order n and, as noted in Remark 8.1, Keum [Ke, Main Theorem] proved that n is the order of some automorphism of a K3 surface in characteristic p if and only if $n \in S_{\mu }'(p)$ (equivalently $n \in S_{\mu }(0)$ ).

If $n \in S_{\mu }'(p)$ and $p \mid n$ , then the examples given in Example 9.6 show that $n \in S_{\mu }(p)$ .

Now, take $n \in S_{\mu }(p)$ with $p \mid n$ , and we will show $n \in S_{\mu }'(p)$ . Write $n = p^e r$ with $p \nmid r$ . Since a smooth K3 surface never admits a $\mu _p$ -action, an example X must have an RDP w. Since $\mu _{p^e}$ -fixed RDPs can be blown up, we may assume w is not $\mu _{p^e}$ -fixed. Note that such RDPs are classified in Proposition 4.12. We show in each case that n belongs to $S_{\mu }'(p)$ .

Suppose w is $D_m$ or $E_m$ . Let $\mu _{p^f s} = \operatorname {\mathrm {Stab}}(w) \subset \mu _{p^e r}$ (with $p \nmid s$ ). Then the pair $(w, p^e s)$ appears in Table 5 and we have $(r/s)m < 22$ . Then we observe that $n \in S_{\mu }'(p)$ except in the following cases: $(p,n,s,(r/s)w) = (2,54,9,3E_7^0), (2,40,5,D_{21}^0), (2,34,17,D_{18}^0), (2,34,17,D_{19}^0)$ . These exceptional cases do not occur, since it follows that $\mu _s$ acts trivially on the classes of every ( $21, 21, 18, 19$ ) exceptional curves, which gives a too large invariant subspace of $H_{\mathrm {\acute et}}^2(\tilde {X}, \mathbb Q_l)$ for an order s automorphism (which must act on $H_{\mathrm {\acute et}}^2$ faithfully with a characteristic polynomial with coefficients in $\mathbb Q$ by Lemma 8.3(1)).

Suppose w is $A_{m'-1}$ . Let $\mu _{p^f s} = \operatorname {\mathrm {Stab}}(w) \subset \mu _{p^e r}$ (with $p \nmid s$ ). We have $0 \leq f < e$ . It follows from Proposition 4.12 that $p^{e-f} \mid m'$ , so write $m' = p^{e-f} m$ with $m \geq 1$ , and that $\mu _{p^f t}$ acts on w symplectically where either $s = t$ or $(s,t) = (2,1)$ . Then $X/\mu _{p^f t}$ has $(r/s) A_{p^f t m' - 1}$ . We have $22> (r/s) (p^f t m' - 1) = (r/s)(p^e t m - 1) \geq r (t/s) m (p^e - 1)$ . If $s = t$ , then this implies $(p^e - 1)r \leq (p^e - 1)rm < 22$ , and if $(s,t) = (2,1)$ , then this implies ( $p \neq 2$ and) $2 \mid r$ and $(p^e - 1)r \leq (p^e - 1)rm < 44$ . We observe that this condition implies either $n \in S_{\mu }'(p)$ or $(p,n) = (5,40), (3,48), (2,34)$ . It remains to show that each of the latter three cases is impossible.

If $(p,n) = (3,48)$ , then $(m,s,t) = (1,2,1)$ . Since $\mu _s$ does not act symplectically, the generator of $\mu _{16}$ acts on $H^0(\tilde {X}, \Omega ^2)$ by a primitive $16$ th root of unity. If $\tilde {X}$ is of finite height, then by Lemma 8.3(3), we have $\operatorname {\mathrm {rank}} T(\tilde {X}) \geq \phi (16) = 8$ and we have $\rho (\tilde {X})> 8 \cdot 2 = 16$ (from $8A_2$ ), contradicting $\rho + \operatorname {\mathrm {rank}} T(\tilde {X}) = 22$ . By the remark after Lemma 8.3, $\tilde {X}$ cannot be supersingular.

If $(p,n) = (2,34)$ , then $(m,s,t) = (1,1,1)$ . Let $e_i$ ( $i \in \mathbb Z/17\mathbb Z$ ) be the exceptional curves above the $\mu _{17}$ -orbit of w, numbered in a way that a generator $g \in \mu _{17}$ acts by $g(e_i) = e_{i+1}$ . Let $L \subset \operatorname {\mathrm {Pic}}(\tilde {X})$ be the sublattice generated by $e_i$ ’s, and $L' = \operatorname {\mathrm {Pic}}(\tilde {X}) \cap \mathbb Q L$ its primitive closure.

First, suppose $\tilde {X}$ is of finite height. Then, by Corollary 6.4 the $\mu _2$ -action is symplectic. We may assume that X is maximal, in which case the number of RDPs on X (which are all of type $A_1$ ) is $8$ by Proposition 6.8, which is not compatible with the $\mu _{17}$ -action.

Next, suppose $\tilde {X}$ is supersingular. By Lemma 8.3(5), the only possible Artin invariant is $\sigma _0 = 4$ . By Lemma 8.4 (applied to $L_1 = L'$ , $L_2 = L^{\prime \perp }$ , $M = \operatorname {\mathrm {Pic}}(\tilde {X})$ , $\bar {M} = \operatorname {\mathrm {Pic}}(\tilde {X})^*$ , and $\bar {L}_i = L_i^*$ ), we have

$$ \begin{align*} \operatorname{\mathrm{rank}} (\operatorname{\mathrm{disc}}(L')) &\leq \dim_{\mathbb F_2}(\operatorname{\mathrm{disc}}(\operatorname{\mathrm{Pic}}(\tilde{X}))) + \operatorname{\mathrm{rank}}(\operatorname{\mathrm{disc}}(L^{\prime \perp})) \\ &\leq \dim_{\mathbb F_2}(\operatorname{\mathrm{disc}}(\operatorname{\mathrm{Pic}}(\tilde{X}))) + \operatorname{\mathrm{rank}}(L^{\prime \perp}) \\ &= 2 \sigma_0 + (22 - \operatorname{\mathrm{rank}} L') = 13. \end{align*} $$

Since $\operatorname {\mathrm {disc}}(L) \cong (\mathbb Z/2\mathbb Z)^{\oplus 17}$ has rank $17$ , we obtain $L \subsetneq L'$ . Let $V \subset 2^{\mathbb Z/17\mathbb Z}$ be the set of subsets $S \subset \mathbb Z/17\mathbb Z$ such that $(1/2) \sum _{i \in S} e_i \in L'$ . Then V is naturally a g-stable $\mathbb F_2$ -vector space, and is nonzero, and we can identify it with a nonzero $\mathbb F_2[x]$ -submodule V of $\mathbb F_2[x] / (x^{17} - 1)$ . Clearly, $V = Q(x) \cdot \mathbb F_2[x] / (x^{17} - 1)$ for some $Q(x) \in \mathbb F_2[x]$ dividing $x^{17} - 1$ . Using the factorization $ x^{17} - 1 = (x - 1) F_{17,1}(x) F_{17,2}(x) $ in $\mathbb F_2[x]$ , where

$$ \begin{align*} F_{17,1}(x) &= x^8 + x^7 + x^6 + x^4 + x^2 + x + 1 \quad \text{and} \\ F_{17,2}(x) &= x^8 + x^5 + x^4 + x^3 + 1 \end{align*} $$

are irreducible, it follows that V contains at least one of

$$ \begin{align*} (x^{17} - 1) / (x - 1) &= x^{16} + \dots + 1, \\ (x^{17} - 1) / F_{17,1}(x) &= x^9 + x^8 + x^6 + x^3 + x + 1, \quad \text{or} \\ (x^{17} - 1) / F_{17,2}(x) &= x^9 + x^6 + x^5 + x^4 + x^3 + 1. \end{align*} $$

Hence, there exists a set $S \in V$ with $\# S = 17$ or $\# S = 6$ . But then, $((1/2) \sum _{i \in S} e_i)^2 = (1/2)^2 \cdot \# S \cdot (-2) \not \in 2\mathbb Z$ , contradiction.

If $(p,n) = (5,40)$ , then $(m,s,t) = (1,2,1)$ and there are four non- $\mu _5$ -fixed $A_4$ on which g acts transitively, where g is a fixed generator of $\mu _8 \subset \mu _{40}$ , and $g^4$ is non-symplectic. Moreover, $\operatorname {\mathrm {Fix}}(g^4)$ is one-dimensional, passing through the four points of type $A_4$ .

By the remark after Lemma 8.3, $\tilde {X}$ cannot be supersingular. This implies that the $\mu _5$ -action is symplectic (Corollary 6.4) and hence the quotient $Y = X/\mu _5$ is an RDP K3 surface.

We will show in a subsequent paper [Mat3, Prop. 2.15(4)] that the $\mu _p$ -action induces a canonical nonzero element $v \in H^0(Y^{\mathrm {sm}}, \Omega ^2) \otimes \operatorname {\mathrm {Der}}(Y)$ , which we can write $v = \omega _Y \otimes D_Y$ (uniquely up to $k^*$ ), such that $D_Y$ is p-closed and satisfies $Y^{D_Y} = {X}^{(p)}$ . It is characterized by $D(f)^p D_Y(h) \omega _Y = d(f^p) \wedge dh$ for local sections f of $\mathcal {O}_X$ and h of $\mathcal {O}_Y$ . Since the $\mu _5$ -action in our case is g-invariant, it follows that $v = \omega _Y \otimes D_Y$ is g-invariant.

Fix a decomposition $v = \omega _Y \otimes D_Y$ . We have $D_Y^p = \phi D_Y$ for some meromorphic function $\phi $ . Since both Y and the quotient $Y^{D_Y} \cong {X}^{(p)}$ are RDP K3 surfaces, it follows from the Rudakov–Shafarevich formula [RS, Cor. 1 to Prop. 3] that $D_Y$ has only isolated fixed points, and this implies that $\phi $ is holomorphic, hence constant. We have $\phi \neq 0$ , since if $\phi = 0$ , then, as we will prove in a subsequent paper [Mat3, Lem. 3.6 or Th. 4.6], the $\alpha _p$ -action corresponding to $D_Y$ must have quotient singularities different from $A_{p-1}$ , but X has only RDPs of type $A_{p-1}$ . Since $\operatorname {\mathrm {Fix}}(D_Y)$ is isolated and $Y^{D_Y} = Y^{g^*(D_Y)}$ , we have $g^*(D_Y) = \lambda D_Y$ with $\lambda \in H^0(Y, \mathcal {O})^* = k^*$ , and since $D_Y^p = \phi D_Y$ with $\phi \in k^*$ , we have $\lambda ^{p-1} = 1$ , and hence $(g^4)^{*}(D_Y) = D_Y$ .

On the other hand, since $\operatorname {\mathrm {Fix}}(g^4 \curvearrowright Y)$ is homeomorphic to $\operatorname {\mathrm {Fix}}(g^4 \curvearrowright X)$ and hence is one-dimensional, we have $(g^4)^{*}(\omega _Y) = - \omega _Y$ . This contradicts the g-invariance of $v = \omega _Y \otimes D_Y$ .

Lemma 8.4. Let $L_1 \subset \bar {L}_1$ , $L_2 \subset \bar {L}_2$ , and $L_1 \oplus L_2 \subset M \subset \bar {M} \subset \bar {L}_1 \oplus \bar {L}_2$ be sequences of abelian groups, where the bars have no specific meaning. Assume that the projection $\bar {M} \to \bar {L}_1$ are surjective and that $M \cap (\bar {L}_1 \oplus 0) = L_1 \oplus 0$ . Then we have $\operatorname {\mathrm {rank}}(\bar {L}_1 / L_1) \leq \operatorname {\mathrm {rank}}(\bar {M}/M) + \operatorname {\mathrm {rank}}(\bar {L}_2 / L_2)$ , where the rank of an abelian group is the minimum number of generators.

Proof. We may assume $L_i = 0$ . The assumption then implies that $M \to \bar {L}_2$ is injective. Since the rank behaves subadditively with respect to subgroups, quotients, and extensions, we obtain $\operatorname {\mathrm {rank}}(\bar {L}_1) \leq \operatorname {\mathrm {rank}}(\bar {M}) \leq \operatorname {\mathrm {rank}}(M) + \operatorname {\mathrm {rank}}(\bar {M}/M) \leq \operatorname {\mathrm {rank}}(\bar {L}_2) + \operatorname {\mathrm {rank}}(\bar {M}/M)$ .

9 Examples

For a projective variety with projective coordinates $(x_i)$ , we use the notation $\operatorname {\mathrm {wt}}(x_i) = (n_i)$ to mean that $\operatorname {\mathrm {wt}}(x_j/x_i) = (n_j - n_i)$ on the affine piece $(x_i \neq 0)$ for each i. Note that $\operatorname {\mathrm {wt}}(x_i) = (n_i)$ is equivalent to $\operatorname {\mathrm {wt}}(x_i) = (a + n_i)$ . We use a similar notation for subvarieties of $\mathbb P(3,1,1,1)$ .

9.1 Symplectic actions

Example 9.1 (Symplectic $\mu _4 \times \mu _4$ -action)

The quartic surface $X = (w^4 + x^4 + y^4 + z^4 + wxyz = 0)$ in characteristic $p = 2$ is an RDP K3 surface. It has six RDPs, all of type $A_3$ , at the points where two of $w,x,y,z$ are $0$ and the others are $1$ . This surface admits a symplectic action of $G = H_1 \times H_2$ , where $H_1 = \mu _4$ and $H_2 = \mu _4$ act by $\operatorname {\mathrm {wt}}(w,x,y,z) = (0,0,1,-1)$ and $\operatorname {\mathrm {wt}}(w,x,y,z) = (0,1,0,-1)$ , respectively.

With respect to the action of the subgroup scheme $\mu _2 \subset H_1$ , the two RDPs at $(0,0,1,1)$ and $(1,1,0,0)$ are fixed and the other four RDPs are non-fixed and non-maximal. The quotient surface by this $\mu _2$ -action is $(W^2 + X^2 + Y^2 + Z^2 + AB = WX - A^2 = YZ - B^2 = 0)$ in $\mathbb P^5$ , where $W = w^2, \dots , Z = z^2$ and $A = wx, B = yz$ , with two RDPs of type $A_7$ at $(Y = Z = B = W+X = W+A = 0),(W = X = A = Y+Z = Y+B = 0)$ and four of type $A_1$ at $(WX = A = YZ = B = W + X + Y + Z = 0)$ .

The quotient morphism by the subgroup scheme $\mu _2 \times \mu _2$ (resp. the full group G) is the relative Frobenius morphism $X \to X^{(2)}$ (resp. $X \to X^{(4)}$ ).

Example 9.2 (Symplectic $\mu _3 \times \mu _3$ -action)

The surface $X = (v^3 + w^3 + x^3 + y^3 + z^3 + vwx = v^2 - yz = 0) \subset \mathbb P^4$ in characteristic $p = 3$ is an RDP K3 surface, and has $2A_5$ at $(1,0,0,1,1),(0,1,-1,0,0)$ and $4A_2$ at $(0,1,0,-1,0),(0,1,0,0,-1),(0,0,1,-1,0),(0,0,1,0,-1)$ . This surface admits a symplectic action of $G = H_1 \times H_2$ , where $H_1 = \mu _3$ and $H_2 = \mu _3$ act by $\operatorname {\mathrm {wt}}(v,w,x,y,z) = (0,1,-1,0,0)$ and $\operatorname {\mathrm {wt}}(v,w,x,y,z) = (0,0,0,1,-1)$ , respectively. Let $D_1,D_2$ be the corresponding derivations. The fixed points of $D_1$ (resp. $D_2$ ) is the first (resp. second) $A_5$ point. The fixed points of $D_1 + D_2$ (resp. $D_1 - D_2$ ), which corresponds to the diagonal (resp. anti-diagonal) subgroup of G, are the first and the fourth (resp. the second and the third) $A_2$ points. The quotient morphism by G is the relative Frobenius morphism $X \to X^{(3)}$ .

Example 9.3 (Symplectic $\mu _n$ -action ( $n = 5,6,7,8$ ))

For each $n = 5,6,7,8$ , let F be a linear combination of the monomials listed in Table 8, in characteristic p, and then $X = (F = 0) \subset \mathbb P^3$ admits a $\mu _n$ -action with the indicated weights. If F is a generic such polynomial, then X is an RDP K3 surface and the $\mu _n$ -action is symplectic. The fixed locus is $X \cap \{(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)\}$ .

Table 8 Examples of symplectic $\mu _n$ -actions on RDP K3 surfaces.

For example, for $n = 5,6,7,8$ , respectively, the polynomials with coefficients $(1,1,1,1,0,0,0)$ , $(1,1,1,1,1,0,0)$ , $(1,1,1,1,1)$ , $(1,1,1,1,1)$ satisfy the condition.

9.2 Non-symplectic actions

Example 9.4 (Non-symplectic $\mu _2$ -action with Enriques quotient in characteristic $2$ )

Following [BM1, §3], let $L_1, L_2, L_3$ be three linear polynomials in $12$ variables, and let $X \subset \mathbb P^5$ be the intersection of three quadrics $F_1, F_2, F_3$ defined by $F_h = L_h(x_k^2, x_i x_j, y_k^2, y_i x_j + x_i y_j + y_i y_j)_{1 \leq k \leq 3, 1 \leq i < j \leq 3} \in k[x_1,x_2,x_3,y_1,y_2,y_3]$ . Then, for generic $L_h$ , X is an RDP K3 surface (with $12$ RDPs of type $A_1$ ), $\mu _2$ acts on ( $\mathbb P^5$ and) X by $\operatorname {\mathrm {wt}}(x_i, y_i + x_i) = (0,1)$ without any fixed point on X, and the quotient $X/\mu _2$ is an Enriques surface.

Example 9.5 (Non-symplectic $\mu _2$ -action with rational quotient in characteristic $2$ )

The quartic surface $w^2 (xy + z^2) + x^4 + y^4 + z^4 + yz(y^2 + z^2) = 0$ is an RDP K3 surface, and the $\mu _2$ -action with $\operatorname {\mathrm {wt}}(w,x,y,z) = (0,1,1,1)$ is non-symplectic. The fixed locus consists of the curve $(w = 0)$ and the RDP $(x = y = z = 0)$ of type $A_1$ . The image of this RDP in the quotient surface is a non-RDP singularity.

In the following example, for two polynomials $A(t),B(t)$ with $\deg A \leq 8$ and $\deg B \leq 12$ , the elliptic (or quasi-elliptic) surface defined by the equation $y^2 = x^3 + A(t) x + B(t)$ is an abbreviation for the projective surface that is the union of four affine surfaces

$$ \begin{align*} &\operatorname{\mathrm{Spec}} k[x,y,t] / (- y^2 + x^3 + A(t) x + B(t)), \\ &\operatorname{\mathrm{Spec}} k[x',y',t^{-1}] / (- y^{\prime 2} + x^{\prime 3} + t^{-8} A(t) x' + t^{-12} B(t)), \\ &\operatorname{\mathrm{Spec}} k[z,w,t] / (- z + w^3 + A(t) wz^2 + B(t) z^3), \\ &\operatorname{\mathrm{Spec}} k[z',w',t^{-1}] / (- z' + w^{\prime 3} + t^{-8} A(t) w'z^{\prime 2} + t^{-12} B(t) z^{\prime 3}), \end{align*} $$

glued by the relations $x' = t^{-4} x$ , $y' = t^{-6} y$ , $z = y^{-1}$ , $w = x y^{-1}$ , $z' = y^{\prime -1} = t^6 y^{-1}$ , and $w' = x' y^{\prime -1} = t^2 x y^{-1}$ . For generic A and B, this is an RDP K3 surface.

Example 9.6 (Non-symplectic $\mu _{n}$ -actions)

Table 9 proves the existence part of Theorem 8.2 for n divisible by p. The first group consists of elliptic (or quasi-elliptic) RDP K3 surfaces, the second of double sextics, and the third of quartics. Only the non- $\mu _n$ -fixed RDPs are listed, except in the example for $(p,n) = (2,32)$ , the $D_{20}^0$ point is fixed and after blowing-up this point, we find a non-fixed $D_{18}^0$ point.

Table 9 Examples of non-symplectic $\mu _n$ -actions on RDP K3 surfaces.

The examples are characteristic p reductions of the examples (of an automorphism of order n) in characteristic $0$ obtained, respectively, by Brandhorst [B, Th. 5.9], Keum [Ke, Exam. 3.2], Kondo [Ko, §§3 and 7], and Oguiso [O1, Prop. 2], except that for the ones marked “?” we could not find a reference. An asterisk means that we made a coordinate change $t \mapsto t^{-1}$ .

9.3 $\mu _p$ -actions on abelian surfaces

As shown in Theorem 6.3, the nontrivial $\mu _p$ -actions of abelian surfaces A, up to automorphisms of $\mu _p$ , are precisely the translations by subgroup schemes of $A[p]$ isomorphic to $\mu _p$ .

Remark 9.7. In the case of finite order automorphisms on abelian surfaces, there are examples with non-abelian quotients. Kummer surfaces in characteristic $\neq 2$ are the minimal resolution of the RDP K3 quotient (with $16A_1$ ) by the symplectic involution $x \mapsto -x$ on abelian surfaces (for characteristic $2$ , see Remark 3.6). Furthermore, certain non-symplectic (or sometimes symplectic) actions give (quasi-)hyperelliptic quotients. It seems that there are no $\mu _p$ -analogue of these actions.

Remark 9.8. If we consider rational vector fields (i.e., possibly with poles) of multiplicative type, there are other kinds of examples. See [KT, Exam. 6.2] for a rational vector field of multiplicative type on an abelian surface (in characteristic $2$ ) with a general type quotient.