Delta-invariants of complete intersection log del Pezzo surfaces

Abstract

We show that complete intersection log del Pezzo surfaces with amplitude one in weighted projective spaces are uniformly $K$-stable. As a result, they admit an orbifold Kähler–Einstein metric.

1. Introduction

Throughout the article, the ground field is assumed to be the field of complex numbers. Let $S$ be a codimension $c$ complete intersection of type $(d_1,\, \ldots,\, d_c)$ in a weighted projective space $\mathbb {P}(a_0,\, \ldots,\, a_n)$ that is quasi-smooth, well-formed and $a_0\leq a_1\leq \cdots \leq a_n < d_1\leq \cdots \leq d_c$. Suppose that $S$ is a log del Pezzo surface. Then we have exactly two possibilities:

1. (A) Either $n=3$ and $S\subset \mathbb {P}(a_0,\,a_1,\,a_2,\,a_3)$ is a hypersurface of degree

   \[ d< a_0+a_1+a_2+a_3 \]
   with amplitude $I=a_0+a_1+a_2+a_3-d$

2. (B) Or $n=4$ and $S\subset \mathbb {P}(a_0,\,a_1,\,a_2,\,a_3,\,a_4)$ is a complete intersection of two hypersurfaces of degrees $d_1$ and $d_2$ such that

   \[ d_1+d_2< a_0+a_1+a_2+a_3+a_4 \]
   with amplitude $I=a_0+a_1+a_2+a_3+a_4-d_1-d_2$.

In the case (A), Johnson and Kollár [9] found the complete list of all possibilities for the quintuple $(a_0,\,a_1,\,a_2,\,a_3,\,d)$ in the case when the amplitude $I$ is one. Moreover, they computed the $\alpha$-invariants and proved the existence of the orbifold Kähler–Einstein metrics in the case when the quintuple $(a_0,\, a_1,\, a_2,\, a_3,\, d)$ is not one of the following four quintuples

\[ (1,2,3,5,10),\quad (1,3,5,7,15),\ (1,3,5,8,16),\ (2,3,5,9,18). \]

To prove the above statement they used the criterion that a log del Pezzo surface $S$ admits an orbifold Kähler–Einstein metric whenever the $\alpha$-invariant of $S$ is bigger than $\frac {2}{3}$. Later, Araujo [1] computed the $\alpha$-invariants for two of these four cases to show the existence of an orbifold Kähler–Einstein metric when $(a_0,\, a_1,\, a_2,\, a_3,\, d) = (1,\, 2,\, 3,\, 5,\, 10)$ or $(a_0,\, a_1,\, a_2,\, a_3,\, d) = (1,\, 3,\, 5,\, 7,\, 15)$ and the defining equation contains the monomial $yzt$ where $x$, $y$, $z$ and $t$ are coordinates with weights $\operatorname {wt}(x) = a_0$, $\operatorname {wt}(y) = a_1$, $\operatorname {wt}(z) = a_2$ and $\operatorname {wt}(t) = a_3$. Finally, Cheltsov, Park and Shramov [2] computed the $\alpha$-invariants for the remaining families.

For the case (A) every log del Pezzo surface $S$ admits an orbifold Kähler–Einstein metric except possibly the case when $(a_0,\,a_1,\,a_2,\,a_3,\,d) = (1,\,3,\,5,\,7,\,15)$ and the defining equation does not contain the monomial $yzt$ whose $\alpha$-invariant is $\frac {8}{15}(<\tfrac {2}{3})$.

Recently Fujita and Odaka introduced $\delta$-invariant which gives a strong criterion showing the uniform $K$-stability of ${{\mathbb {Q}}}$-Fano varieties (see [8]).

Theorem 1.1 Let $X$ be a $\mathbb {Q}$-Fano variety. Then X is uniformly $K$-stable if and only if $\delta (X) > 1$.

The estimation of the $\delta$-invariant has been investigated on several log del Pezzo surfaces in [4–7, 14, 15]. Moreover Li, Tian and Wang generalized in [13] the result of Chen, Donaldson, Sun and Tian for the $K$-polystability and the existence of the Kähler–Einstein metric to some singular Fano varieties. In virtue of the $\delta$-invariant method and the result [13], the paper [3] completes the problem of the existence of the (orbifold) Kähler–Einstein metric on del Pezzo hypersurfaces with $I=1$, case (A):

Theorem 1.2 [3]

Let $S$ be a quasi-smooth hypersurface in ${{\mathbb {P}}}(1,\,3,\,5,\,7)$ of degree $15$ such that its defining equation does not contain $yzt$. Then the surface $S$ admits an orbifold Kähler–Einstein metric.

Corollary 1.3 Every quasi-smooth hypersurface with $I=1$ admits an orbifold Kähler-Einstein metric.

In [10] and [11], we classified the log del Pezzo surfaces $S$ for the case (B) when $S\subset {{\mathbb {P}}}(a_0,\, a_1,\, a_2,\, a_3,\, a_4)$ are quasi-smooth and well-formed complete intersection log del Pezzo surfaces given by two quasi-homogeneous polynomials of degrees $d_1$ and $d_2$ with amplitude $1$, and not being the intersection of a linear cone with another hypersurface. Then there are 42 families. We denote family No. $i$ as the number $i$ in the first column $\Gamma$ of the table which is represented in [11, section 5].

Suppose that the log del Pezzo surface $S$ is not one of the following:

• • No. 3 : a complete intersection of two hypersurfaces of degrees $6$ and $8$ embedded in ${{\mathbb {P}}}(1,\,2,\,3,\,4,\,5)$ such that the defining equation of the hypersurface of degree 6 does not contain the monomial $yt$, where $y$ is the coordinate function of weight $2$ and $t$ is the coordinate function of weight $4$.

• • No. 40 : a complete intersection of two hypersurfaces of degree $2n$ embedded in ${{\mathbb {P}}}(1 ,\,1,\, n,\, n,\, 2n-1)$ where $n$ is a positive integer.

Then the $\alpha$-invariant of $S$ is bigger than $\tfrac {2}{3}$, in fact they are bigger or equal to one, so that it admits an orbifold Kähler–Einstein metric (see [10, theorem 1.9] and [11, theorem 1.2]).

The present article completes the existence of the orbifold Kähler–Einstein metric of the remaining two cases.

Theorem 1.4 Let $S$ be a quasi-smooth member of family No. $i$ with $i\in \{3,\,40\}$. Then the log del Pezzo surface $S$ is uniformly $K$-stable so that it admits an orbifold Kähler–Einstein metric.

Corollary 1.5 Every quasi-smooth weighted complete intersection with $I=1$ admits an orbifold Kähler–Einstein metric.

2. Preliminary

2.1 Notation

Throughout the paper we use the following notations:

• • For positive integers $a_0$, $a_1$, $a_2$, $a_3$ and $a_4$, ${{\mathbb {P}}}(a_0,\,a_1,\,a_2,\,a_3,\,a_4)$ is the weighted projective space. We assume that $a_0\leq a_1\leq a_2\leq a_3\leq a_4$.

• • We usually write $x$, $y$, $z$, $t$ and $w$ for the weighted homogeneous coordinates of ${{\mathbb {P}}}(a_0,\,a_1,\,a_2,\,a_3,\,a_4)$ with weights $\operatorname {wt}(x)=a_0$, $\operatorname {wt}(y)=a_1$, $\operatorname {wt}(z)=a_2$, ${\operatorname {wt}(t)=a_3}$ and $\operatorname {wt}(w)=a_4$.

• • $S\subset {{\mathbb {P}}}(a_0,\,a_1,\,a_2,\,a_3,\,a_4)$ denotes a quasi-smooth complete intersection log del Pezzo surface given by quasi-homogeneous polynomials of degrees $d_1$ and $d_2$.

• • The integer $I = a_0 + a_1 + a_2 + a_3 + a_4 - d_1 - d_2$ is called the amplitude of $S$.

• • $H_{*}$ is the hyperplane section on the log del Pezzo surface $S$ cut out by the equation $* = 0$.

• • $\mathsf p_x$ denotes the point on $S$ given by $y=z=t=w=0$. The points $\mathsf p_y$, $\mathsf p_z$, $\mathsf p_t$ and $\mathsf p_w$ are defined in a similar way.

• • $-K_S$ denotes the anti-canonical divisor of $S$.

2.2 Foundation

$X$ is ${{\mathbb {Q}}}$-Fano variety, i.e., a normal projective ${{\mathbb {Q}}}$-factorial variety with at most terminal singularities such that $-K_X$ is ample.

Definition 2.1 Let $(X,\,D)$ be a pair, that is, $D$ is an effective ${{\mathbb {Q}}}$-divisor, and let $\mathsf p \in X$ be a point. We define the log canonical threshold (LCT, for short) of $(X,\,D)$ and the log canonical threshold of $(X,\,D)$ at $\mathsf p$ to be the numbers

\begin{align*} \operatorname{lct} (X,D) & = \sup \{\, c \mid (X, c D)\text{ is log canonical}\}, \\ \operatorname{lct}_{\mathsf p} (X,D) & = \sup \{\, c \mid (X, c D)\text{ is log canonical at }\mathsf p\}, \end{align*}

respectively. We define

\[ \operatorname{lct}_{\mathsf p} (X) = \inf \{\, \operatorname{lct}_{\mathsf p} (X,D) \mid D \text{ is an effective }{{\mathbb{Q}}}\text{-divisor}, D \equiv{-}K_X\}, \]

and for a subset $\Sigma \subset X$, we define

\[ \operatorname{lct}_{\Sigma} (X) = \inf \{\operatorname{lct}_{\mathsf p} (X) \mid \mathsf p \in \Sigma\}. \]

The number $\alpha (X) := \operatorname {lct}_X (X)$ is called the global log canonical threshold (GLCT, for short) or the $\alpha$-invariant of $X$

Let $S$ be a surface with at most cyclic quotient singularities, and let $D$ be an effective ${{\mathbb {Q}}}$-divisor on $X$.

Lemma 2.2 [12]

Let $\mathsf p$ be a smooth point of $S$. Suppose that the log pair $(S,\, D)$ is not log canonical at the point $\mathsf p$. Then $\operatorname {mult}_{\mathsf p}(D) > 1$.

Suppose that $S$ has a cyclic quotient singular point $\mathsf q$ of type $\frac {1}{r}(a,\,b)$. Then there is an orbifold chart $\pi \colon \bar {U}\to U$ for some open set $\mathsf q\in U$ on $S$ such that $\bar {U}$ is smooth and $\pi$ is a cyclic cover of degree $r$ branched over $\mathsf q$.

Lemma 2.3 [12]

Let $\bar {\mathsf q}\in \bar {U}$ be the point such that $\pi (\bar {\mathsf q}) = \mathsf q$. Then the log pair $(U,\, D|_U)$ is log canonical at the point $\mathsf q$ if and only if the log pair $(\bar {U},\, \bar {D}|_{\bar {U}})$ is log canonical at the point $\bar {\mathsf q}$ where $\bar {D}=\pi ^{*}(D|_{U})$.

Definition 2.4 [8]

Let $k$ be a positive integer. We set $h=h^{0}(S,\,-kK_S)$. Given any basis

\[ s_1,\ldots , s_h \]

of $H^{0}(S,\,-kK_S)$, taking the corresponding divisors $D_1,\,\ldots,\,D_h$ with $D_i\sim -kK_S$, we get an anti-canonical ${{\mathbb {Q}}}$-divisor

\[ D:=\frac{D_1+\ldots+D_h}{kh}. \]

We call this kind of anti-canonical ${{\mathbb {Q}}}$-divisor an anti-canonical ${{\mathbb {Q}}}$-divisor of $k$-basis type.

Then we can define the $\delta$-invariant of $S$ using an anti-canonical ${{\mathbb {Q}}}$-divisor of $k$-basis type. The definition of the $\delta$-invariant of a Fano variety is the following.

Definition 2.5 [8]

For $k\in {{\mathbb {Z}}}_{>0}$, set

\[ \delta_k(S):=\inf\{~\operatorname{lct}(S,D)~|~D \text{ is of } k\text{ -basis type}~ \}. \]

Moreover, we define

\[ \delta(S):=\limsup_{k\to \infty}\delta_k(S). \]

It is called the $\delta$-invariant of $S$.

Definition 2.6 Let $X$ be an irreducible projective variety of dimension $n$, and let $D$ be a Cartier divisor on X. The volume of $D$ is defined to be the non-negative real number

\[ \operatorname{vol}(D)=\operatorname{vol}_X(D)=\limsup_{m\to \infty}\frac{h^{0}(X,\mathcal{O}_X(mD))}{m^{n}/n!}. \]

For a ${{\mathbb {Q}}}$-divisor $D$ on the surface $S$ we can define its volume using the identity

\[ \operatorname{vol}(D) = \frac{\operatorname{vol}(\lambda D)}{\lambda^{2}} \]

for an appropriate positive rational number $\lambda$.

Let $D$ be an anti-canonical ${{\mathbb {Q}}}$-divisor of $k$-basis type with $k\gg 1$, and let $C$ be an irreducible reduced curve on $S$. We write

\[ D = aC + \Delta \]

where $a$ is non-negative real number and $\Delta$ is an effective ${{\mathbb {Q}}}$-divisor such that $C\not \subset \operatorname {Supp}(\Delta )$. Let

\[ \tau = \sup\{~x\in{{\mathbb{R}}}_{> 0}~|~D - xC \text{ is pseudoeffective}~\}. \]

In the case that $D$ is an ample ${{\mathbb {Q}}}$-divisor of $k$-basis type with $k\gg 1$ we can find a better bound for $a$. One such estimate is given by the following very special case of [8, lemma 2.2].

Theorem 2.7 [3, theorem 2.9]

Suppose that $D$ is a big ${{\mathbb {Q}}}$-divisor of $k$-basis type for $k\gg 1$. Then

\[ a\leq \int_0^{\tau} \operatorname{vol}(D - xC)dx + \epsilon_k \]

where $\epsilon _k$ is a small constant depending on $k$ such that $\epsilon _k\to 0$ as $k\to \infty$.

Corollary 2.8 [3, corollary 2.10]

Suppose that $D$ is a big ${{\mathbb {Q}}}$-divisor of $k$-basis type for $k\gg 0,$ and

\[ C\sim_{{{\mathbb{Q}}}} \mu D \]

for some positive rational number $\mu$. Then

\[ a\leq \frac{1}{3\mu} + \epsilon_k, \]

where $\epsilon _k$ is a small constant depending on $k$ such that $\epsilon _k \to 0$ as $k\to \infty$.

3. Family No. $3$

In this section we prove the following theorem:

Theorem 3.1 Let $S$ be a quasi-smooth member of family No. $3$. Then $\delta (S) \geq \frac {5}{4}$. Moreover, $S$ admits an orbifold Kähler–Einstein metric.

Proof. Let $D$ be an anti-canonical ${{\mathbb {Q}}}$-divisor of $k$-basis type on $S$ with $k\gg 0$. By lemmas 3.2–3.4 the log pair $(S,\, \frac {5}{4}D)$ is log canonical. Therefore $\delta (S)\geq \frac {5}{4}$.

We divide the proof of the above theorem into a sequence of lemmas. Let $S\subset {{\mathbb {P}}}(1,\,2,\,3,\,4,\,5)$ be a quasi-smooth complete intersection log del Pezzo surface given by two quasi-homogeneous polynomials of degrees $6$ and $8$. By suitable coordinate change we may assume that $S$ is given by

\[ \begin{array}{l} wx + \xi ty + z^{2} + y^{3} = 0,\\ wz + t^{2} + g(x,y) = 0, \end{array} \]

where $\xi$ is a constant and $g(x,\,y)$ is a quasi-homogeneous polynomial of degree $8$. Then $S$ is singular only at the point $\mathsf p_w$, which is a cyclic quotient singularity of type $\tfrac {1}{5}(4,\,3)$. Since the defining equation of degree $6$ of a member of family No. $3$ does not contain the monomial $ty$, $\xi = 0$. Thus $S$ is given by

\begin{align*} F & = wx + z^{2} + y^{3} = 0,\\ G & = wz + t^{2} + g(x,y) = 0. \end{align*}

Let $H_x$ be the hyperplane section given by $x = 0$. Then it is isomorphic to the variety embedded in ${{\mathbb {P}}}(2,\,3,\,4,\,5)$ given by

\[ \begin{array}{l} z^{2} + y^{3} = 0,\\ wz + t^{2} + \zeta y^{4} = 0, \end{array} \]

where $\zeta = g(0,\,1)$. We consider the open set $U = S \setminus H_w$ where $H_w$ is the hyperplane section given by $w=0$. $H_x|_U$ is isomorphic to the ${{\mathbb {Z}}}_5$-quotient of the affine curve given by

(3.1)\begin{equation} (t^{2} + \zeta y^{4})^{2} + y^{3} = 0 \end{equation}

in ${{\mathbb {A}}}^{2}$. From the equation (3.1), we can see that $H_x$ is irreduciblyreduced and singular at the point $\mathsf p_w$. Also, we have $\operatorname {lct}(S,\, H_x) = \frac {7}{12}$.

Let $D$ be an anti-canonical ${{\mathbb {Q}}}$-divisor of $k$-basis type on $S$ with $k\gg 0$. We put $\lambda = \frac {5}{4}$.

Lemma 3.2 The log pair $(S,\, \lambda D)$ is log canonical along $H_x\setminus \{\mathsf p_w\}$.

Proof. Suppose that the log pair $(S,\, \lambda D)$ is not log canonical at some point $\mathsf p\in H_x\setminus \{\mathsf p_w\}$. We write

\[ D=aH_x + \Delta \]

where $a$ is non-negative rational number and $\Delta$ is an effective divisor such that $H_x\not \subset \operatorname {Supp}(\Delta )$. By corollary 2.8 we have $a\leq \frac {1}{3} + \epsilon _k < \frac {9}{25}$ for $k \gg 0$. Since $\lambda a\leq 1$ the log pair $(S,\, H_x + \lambda \Delta )$ is not log canonical at the point $\mathsf p$. By the inversion of adjunction formula the log pair $(H_x,\, \lambda \Delta |_{H_x})$ is not log canonical at point $\mathsf p$. We have the inequalities

\[ \frac{1}{\lambda}<\operatorname{mult}_{\mathsf p}(\Delta|_{H_x}) \leq \Delta\cdot H_x = (D - aH_x)\cdot H_x = \frac{2}{5} - \frac{2}{5}a, \]

which imply that $a< -1$. This is impossible. Therefore the log pair $(S,\, \lambda D)$ is log canonical along $H_x\setminus \{\mathsf p_w\}$.

Lemma 3.3 The log pair $(S,\, \lambda D)$ is log canonical long $S\setminus H_x$.

Proof. Suppose that the log pair $(S,\, \lambda D)$ is not log canonical at some point $\mathsf p\in S\setminus H_x$. By suitable coordinate change we can assume that $\mathsf p = \mathsf p_x$.

Let $C$ be the curve on $S$ cut out by the equation $y=0$. Then $C$ passes through the point $\mathsf p$. Since the curve $C$ is smooth at $\mathsf p_w$ and $C\cdot H_x= \frac {4}{5}$, it is irreducible and reduced. Let $\mathcal {L}$ be the pencil cut out by the equations $\alpha xy + \beta z = 0$ where $[\alpha : \beta ]\in {{\mathbb {P}}}^{1}$. The base locus of $\mathcal {L}$ is given by $z=yx=0$. Since $S\cap H_x \cap H_z = \{\mathsf p_y\}$ and $S\cap H_y \cap H_z = \{\mathsf p_x,\, \mathsf p_w\}$ we have $\operatorname {BS}(\mathcal {L}) = \{\mathsf p_x,\, \mathsf p_y,\, \mathsf p_w\}$. Thus there is a general member $M\in \mathcal {L}$ such that $\mathsf p\in M$ and $C\not \subset \operatorname {Supp}(M)$. We have

\[ \operatorname{mult}_{\mathsf p}(M)\operatorname{mult}_{\mathsf p}(C) \leq M\cdot C = \frac{12}{5}. \]

It implies that $\operatorname {mult}_{\mathsf p}(C)$ is either $1$ or $2$. We write

\[ D = bC+\Sigma \]

where $b$ is non-negative rational number and $\Sigma$ is an effective ${{\mathbb {Q}}}$-divisor such that $C\not \subset \operatorname {Supp}(\Sigma )$. By Corollary 2.8, we have $b\leq \frac {1}{6} + \epsilon _k < \frac {1}{3}$ for $k \gg 0$.

We assume that $\operatorname {mult}_{\mathsf p}(C) = 1$. Since $\lambda b\leq 1$ the log pair $(S,\, C + \lambda \Sigma )$ is not log canonical at the point $\mathsf p$. By the inversion of adjunction formula the log pair $(C,\, \lambda \Sigma |_C)$ is not log canonical at the point $\mathsf p$. We have the inequalities

\[ \frac{1}{\lambda}<\operatorname{mult}_{\mathsf p}(\Sigma|_C)\leq \Sigma\cdot C = (D - bC)\cdot C = \frac{4}{5} - \frac{8}{5}b. \]

They imply that $b<0$. It is impossible. Thus $\operatorname {mult}_{\mathsf p}(C) = 2$. From lemma 2.2 we have the following inequalities

\[ 2\left(\frac{1}{\lambda} - 2b\right) < \operatorname{mult}_{\mathsf p}(C)\operatorname{mult}_{\mathsf p}(D - bC) \leq C\cdot(D - bC) = \frac{4}{5} - \frac{8}{5}b. \]

Then we have $\frac {1}{3}< b$. It is impossible. Thus the log pair $(S,\, \lambda D)$ is log canonical along $S\setminus H_x$.

Lemma 3.4 The log pair $(S,\, \lambda D)$ is log canonical at $\mathsf p_w$.

Proof. Suppose that the log pair $(S,\, \lambda D)$ is not log canonical at $\mathsf p_w$. We consider the open set $U$ given by $w\neq 0$. Then we may regard $y$ and $t$ are local coordinates with weights $\operatorname {wt}(y) = 4$ and $\operatorname {wt}(t) = 3$ in $U$. Let $\pi \colon \bar {S}\to S$ be the weighted blow-up at $\mathsf p_w$ with weights $\operatorname {wt}(y) = 4$ and $\operatorname {wt}(t) = 3$. Then $\bar {S}$ has the singular points $\mathsf q_1$ and $\mathsf q_2$ of types $\frac {1}{4}(1,\,1)$ and $\frac {1}{3}(1,\,1)$, respectively. We have

\[ K_{\bar{S}} \sim_{{{\mathbb{Q}}}} \pi^{*}(K_S) + \frac{2}{5}E,\quad \bar{H_x} \sim_{{{\mathbb{Q}}}} \pi^{*}(H_x) - \frac{12}{5}E \]

where $\bar {H_x}$ is the strict transform of $H_x$ and $E$ is the exceptional divisor of $\pi$. We write

\[ D = aH_x + \Delta \]

where $a$ is a non-negative rational number and $\Delta$ is an effective ${{\mathbb {Q}}}$-divisor such that $H_x\not \subset \operatorname {Supp}(\Delta )$. By corollary 2.8, we have

(3.2)\begin{equation} a\leq \frac{9}{25} \end{equation}

for $k\gg 0$. We also have

\[ \bar{\Delta} \sim_{{{\mathbb{Q}}}} \pi^{*}(\Delta) - mE \]

where $\bar {\Delta }$ is the strict transform of $\Delta$ and $m$ is a non-negative rational number. To obtain a bound of $m$ we consider the inequality

\[ 0\leq \bar{\Delta}\cdot \bar{H_x} = (\pi^{*}(\Delta) - mE)\cdot \left(\pi^{*}(H_x) - \frac{12}{5}E\right) = \Delta\cdot H_x + \frac{12}{5}mE^{2}. \]

Since $\Delta \cdot H_x = (D - aH_x)\cdot H_x = \frac {2}{5} - \frac {2}{5}a$ and $E^{2} = -\frac {5}{12}$, we have

(3.3)\begin{equation} m\leq \frac{2}{5} - \frac{2}{5}a. \end{equation}

Meanwhile, we have

\[ K_{\bar{S}} + \lambda (a \bar{H_x} + \bar{\Delta}) + \mu E \sim_{{{\mathbb{Q}}}} \pi^{*}(K_S + \lambda D) \]

where

\[ \mu = \lambda\left(\frac{12}{5}a + m \right) - \frac{2}{5}. \]

It implies that the log pair $(\bar {S},\, \lambda (a \bar {H_x} + \bar {\Delta }) + \mu E)$ is not log canonical at some point $\mathsf q\in E$. From the inequalities (3.2) and (3.3) we have $\mu \leq 1$. It implies that the log pair $(\bar {S},\, \lambda (a \bar {H_x} + \bar {\Delta }) +E)$ is not log canonical at the point $\mathsf q$. We consider the case that $E$ is smooth at the point $\mathsf q$. By the inversion of adjunction formula the log pair $(E,\, \lambda (a \bar {H_x} + \bar {\Delta })|_E)$ is not log canonical at $\mathsf q$. If $\mathsf q\not \in \bar {H_x}$ then the log pair $(E,\, \lambda \bar {\Delta }|_E)$ is not log canonical at $\mathsf q$. From this we have the inequalities

\[ \frac{1}{\lambda} < \operatorname{mult}_{\mathsf q}(\bar{\Delta}|_E) \leq \bar{\Delta}\cdot E ={-}mE^{2} = \frac{5}{12}m. \]

They imply that $\frac {48}{25}< m$. From the inequality (3.3), it is impossible. Thus $\mathsf q\in \bar {H_x}$. From lemma 2.2 and the inequality (3.3) we have the inequalities

\[ \frac{1}{\lambda} < \operatorname{mult}_{\mathsf q}((a \bar{H_x} + \bar{\Delta})|_E)\leq (a\bar{H_x} + \bar{\Delta})\cdot E = a + \frac{5}{12}m \leq \frac{1+5a}{6}. \]

They imply that $\frac {19}{25}< a$. From the inequality (3.2), it is impossible. Thus $E$ is singular at the point $\mathsf q$. Also, the point $\mathsf q$ is either $\mathsf q_1$ or $\mathsf q_2$.

Suppose that $\mathsf q = \mathsf q_1$. Then there is a cyclic cover $\varphi \colon \tilde {U}\to \bar {U}$ of degree $4$ branched over $\mathsf q$ for some open set $\mathsf q\in \bar {U}$ on $\bar {S}$ such that $\tilde {U}$ is smooth. From lemma 2.3, the log pair $(\tilde {U},\, \lambda \tilde {\Delta } + \tilde {E})$ is not log canonical at some point $\tilde {\mathsf q}$ where $\tilde {\Delta } = \varphi ^{*}(\Delta |_U)$, $\tilde {E} = \varphi ^{*}(E|_U)$ and $\varphi (\tilde {\mathsf q}) = \mathsf q$. By the inversion of adjunction formula the log pair $(\tilde {E},\, \lambda \tilde {\Delta }|_{\tilde {E}})$ is not log canonical at the point $\tilde {\mathsf q}$. From this we have the inequalities

\[ \frac{1}{\lambda} < \operatorname{mult}_{\tilde{\mathsf q}}(\tilde{\Delta}|_{\tilde{E}}) \leq 4\bar{\Delta}\cdot E ={-}4mE^{2} = \frac{5}{3}m. \]

They imply that $\frac {12}{25} < m$. From the inequality (3.3), it is impossible. Thus $\mathsf q = \mathsf q_2$. Similarly, we can see that this case is impossible. Therefore the log pair $(S,\, \lambda D)$ is log canonical at the point $\mathsf p_w$.

By the above lemmas we prove that the log pair $(S,\,\lambda D)$ is log canonical.

4. On smooth points of family No. $40$

Let $S_n\subset {{\mathbb {P}}}(1,\,1,\,n,\,n,\,2n-1)$ be a quasi-smooth complete intersection log del Pezzo surface given by two quasi-homogeneous polynomials of degree $2n$, where $n$ is a positive integer bigger than $1$. By suitable coordinate change we may assume that $S_n$ is given by

\[ \begin{array}{l} wx + z^{2} + zf_n(x,y) + t\hat{f}_n(x,y) + f_{2n}(x,y) = 0,\\ wy + t^{2} + zg_n(x,y) + t\hat{g}_n(x,y) + g_{2n}(x,y) = 0 \end{array} \]

where $f_i$, $\hat {f}_i$, $g_i$ and $\hat {g}_i$ are homogeneous polynomials of degree $i$. Then $S_n$ is only singular at the point $\mathsf p_w$ of type $\frac {1}{2n-1}(1,\,1)$. In the paper [10], we have $\alpha (S_2) = 7/10$. It implies that $S_2$ admits an orbifold Kähler–Einstein metric. Thus we only consider the cases that $n \geq 3$.

Let $D$ be an anti-canonical ${{\mathbb {Q}}}$-divisor of $k$-basis type on $S_n$ with $k\gg 0$. We set $\lambda = \frac {6n}{4n+3}$. To prove that $\delta (S_n) > 1$ along the smooth points of $S_n$, we consider the following.

Lemma 4.1 The log pair $(S_n,\, \lambda D)$ is log canonical along $S_n\setminus \{\mathsf p_w\}$

Proof. For the convenience, we set $S = S_n$. Suppose that the log pair $(S,\, \lambda D)$ is not log canonical at some point $\mathsf p\in S\setminus \{\mathsf p_w\}$. Let $\mathcal {L}=|-K_S|$ be the pencil cut out on $S$ by the equations $\alpha x + \beta y = 0$ where $[\alpha : \beta ]\in {{\mathbb {P}}}^{1}$. Since the point $\mathsf p$ is not the point $\mathsf p_w$, there is the unique curve $C\in \mathcal {L}$ passing through $\mathsf p$. Without loss of generality we can assume that $\mathsf p$ is contained in the open set $U_x$ given by $x = 1$. Then $C$ is given by the equation $y = \xi x$ on $S$ where $\xi$ is a constant. On the open set $U_x$, the affine curve $C|_{U_x}$ is given by

\[ \begin{array}{l} w + z^{2} + zf_n(1, \xi) + t\hat{f}_n(1, \xi) + f_{2n}(1, xi) = 0,\\ \xi w + t^{2} + zg_n(1, \xi) + t\hat{g}_n(1, \xi) + g_{2n}(1, \xi) = 0 \end{array} \]

Thus it is isomorphic to the variety given by

(4.1)\begin{equation} \xi_1 z^{2} + t^{2} + \xi_2 z + \xi_3 t + \xi_4 = 0 \end{equation}

where $\xi _1\ldots,\,\xi _4$ are constants. Since $S$ is quasi-smooth at least one $\xi _i$ in $i\in \{1,\,2,\,3,\,4\}$ is non-zero. It implies that the rank of the quadratic equation (4.1) is either $1$ or $2$. We assume that $C$ is irreducible. By the quadratic equation (4.1), $C$ is smooth at the point $\mathsf p$. We write

\[ D=aC+\Delta \]

where $\Delta$ is an effective ${{\mathbb {Q}}}$-divisor such that $C\not \subset \operatorname {Supp}(\Delta )$ and $a$ is a non-negative constant. By corollary 2.8 we have $\lambda a \leq 1$. By the inversion of adjunction formula, the log pair $(C,\, \lambda \Delta |_C)$ is not log canonical at $\mathsf p$. Then we have the inequalities

\[ \frac{1}{\lambda} < \operatorname{mult}_{\mathsf p}(\Delta|_C) \leq \Delta\cdot C = \frac{4}{2n-1} - \frac{4a}{2n-1}. \]

The above inequalities imply that $a$ is negative. This is impossible. Thus $C$ is reducible. We now turn to the case that $C$ is the sum of two irreducible curves $L_1$ and $L_2$, that is, we write

\[ C = L_1 + L_2. \]

Then $L_1$ and $L_2$ satisfy the following intersection numbers:

\[ L_1\cdot ({-}K_S) = L_2\cdot ({-}K_S) = \frac{2}{2n-1},\quad L_1\cdot L_2 = \frac{2n}{2n-1},\quad L_1^{2} = L_2^{2} ={-}\frac{2n-2}{2n-1}. \]

Without loss of generality we can assume that $\mathsf p\in L_1$. We write

\[ D=bL_1 + \Sigma \]

where $\Sigma$ is an effective ${{\mathbb {Q}}}$-divisor such that $L_1\not \subset \operatorname {Supp}(\Sigma )$ and $b$ is a non-negative number. By theorem 2.7, we have

\[ b\leq\frac{1}{D^{2}}\int_0^{\tau(L_1)} \operatorname{vol}(D-xL_1)dx + \epsilon_k \]

where $\epsilon _k$ is a small constant depending on $k$ such that $\epsilon _k\to 0$ as $k\to \infty$. Since

\[ D - xL_1\sim_{{{\mathbb{Q}}}} (1-x)L_1+L_2 \]

and $L_2^{2} < 0$, we have $\operatorname {vol}(D-xL_1)=0$ for $x\geq 1$. It implies that $\tau (L_1) = 1$. Meanwhile, the equalities

\[ (D - xL_1)\cdot L_2 = \left((1-x)L_1 + L_2\right)\cdot L_2 = \frac{2}{2n-1} - \frac{2n}{2n-1}x \]

imply that $(D - xL_1)$ is nef whenever $\frac {1}{n}\geq x$. Thus

\[ \operatorname{vol}(D - xL_1) = (D - xL_1)^{2} = \frac{4}{2n-1} - \frac{4}{2n-1}x -\frac{2n-2}{2n-1}x^{2} \]

for $\frac {1}{n}\geq x$. We next consider the volume of $D - xL_1$ for $1\geq x \geq \frac {1}{n}$. Let

\[ P = (1-x)D + (1-x)\frac{1}{n-1}L_2 \]

be the nef divisor for $1\geq x \geq \frac {1}{n}$. Then we write

\[ D - xL_1 = P + \left(\frac{n}{n-1}x - \frac{1}{n-1}\right)L_2. \]

Since $P\cdot L_2 = 0$, the right-hand side of the above equation is the Zariski decomposition of $D - xL_1$. Thus

\[ \operatorname{vol}(D - xL_1) = P^{2} = \frac{2}{n-1}(1-x)^{2} \]

for $1\geq x \geq \frac {1}{n}$. Then we have

\begin{align*} & \displaystyle\frac{1}{D^{2}}\int_0^{\tau(L_1)} \operatorname{vol}(D-xL_1)dx \\ & \quad=\displaystyle\frac{2n-1}{4}\left(\int_{0}^{\frac{1}{n}}\frac{4}{2n-1} - \frac{4}{2n-1}x -\frac{2n-2}{2n-1}x^{2} dx + \int_{\frac{1}{n}}^{1} \frac{2}{n-1}(1-x)^{2} dx\right)\\ & \quad= \displaystyle\frac{2n-1}{4}\left(\frac{12n^{2}-8n+2}{3(2n-1)n^{3}} + \frac{2(n-1)^{2}}{3n^{3}}\right) = \frac{2n+1}{6n}. \end{align*}

Thus we obtain

\[ b\leq \frac{2n+1}{6n} + \epsilon_k. \]

It implies that $\lambda b \leq 1$. By the inversion of adjunction formula we have

\[ \frac{1}{\lambda} < (D - bL_1)\cdot L_1 = \frac{2}{2n-1} + \frac{2n-2}{2n-1}b. \]

It implies that

\[ \frac{(2n-3)(4n+1)}{6n(2n-2)} = \left(\frac{1}{\lambda} - \frac{2}{2n-1}\right)\frac{2n-1}{2n-2} < b. \]

This is impossible. Therefore the log pair $(S,\, \lambda D)$ is log canonical along $S\setminus \{\mathsf p_w\}$.

5. On the singular point of family No. $40$

In this section we prove the following theorem.

Theorem 5.1 Let $S_n\subset {{\mathbb {P}}}(1,\,1,\,n,\,n,\,2n-1)$ be a quasi-smooth member of family No. $40$ where $n$ is a positive integer. Then $\delta (S_n) > \frac {6n}{4n+3}$. Moreover, $S_n$ admits an orbifold Kähler–Einstein metric.

We divide the proof of the above theorem into a sequence of lemmas.

5.1 Basis

Let $\mathcal {L} = H^{0}(S_n,\, \mathcal {O}_{S_n}(k))$ be the vector space where $k$ is a positive integer. In this subsection, we find a monomial basis of $\mathcal {L}$. We define a subset of $\mathcal {L}$ as follows:

\[ \mathcal{B} = \left\{f\in {{\mathbb{C}}}[x,y,z,t,w]_k \middle\vert \begin{array}{l} f \text{ is a monomial whose form is one of the following:}\\ w^{e}, z^{c}t^{d}w^{e}, x^{a}y^{b}t^{d}, x^{a}y^{b}zt^{d} \text{ or } x^{a}z^{c}t^{d}. \end{array} \right\} \]

where ${{\mathbb {C}}}[x,\,y,\,z,\,t,\,w]_k$ is the set of quasi-homogeneous polynomials of degree $k$ with weights $\operatorname {wt}(x) = \operatorname {wt}(y) = 1$, $\operatorname {wt}(z) = \operatorname {wt}(t) = n$ and $\operatorname {wt}(w) = 2n - 1$. The equations

(5.1)\begin{equation} -wx = z^{2} + zf_n(x,y) + t\hat{f}_n(x,y) + f_{2n}(x,y) \end{equation}

and

(5.2)\begin{equation} -wy = t^{2} + zg_n(x,y) + t\hat{g}_n(x,y) + g_{2n}(x,y) \end{equation}

hold in $S_n$. From the equations (5.1) and (5.2), we can obtain

(5.3)\begin{equation} yz^{2} = xt^{2} + zh_{n+1}(x,y) + t\hat{h}_{n+1}(x,y) + h_{2n+1}(x,y). \end{equation}

From the equations (5.1), (5.2) and (5.3) we can see that $\mathcal {L}$ is generated by $\mathcal {B}$ on $S_n$.

Claim. The set $\mathcal {B}$ is the basis of $\mathcal {L}$.

In a neighbourhood $U$ of $S_n$ at $\mathsf p_w$, we may regard $z$ and $t$ are local coordinates with weights $\operatorname {wt}(z) = 1$ and $\operatorname {wt}(t) = 1$. Then $U$ is isomorphic to the quotient of ${{\mathbb {C}}}^{2}$ by the action $\zeta \cdot (z,\, t) \mapsto (\zeta z,\, \zeta t)$ where $\zeta$ is a primitive $(2n-1)$-th root of unity. We have the isomorphism $\sigma \colon {{\mathbb {C}}}/{{\mathbb {Z}}}_{2n-1}\to U$ given by $(z,\,t)\mapsto (z^{2} + f_{>2n},\, t^{2} + g_{> 2n},\,z,\,t)$ where $f_{>2n}$ and $g_{>2n}$ are power series such that the orders are greater than $2n$. Then for a section $s(x,\, y,\, z,\, t,\, w)\in \mathcal {L}$ the local equation in $U$ is given by $\sigma ^{*}(s(x,\, y,\, z,\, t,\, 1))$. We consider the following set:

\[ \mathcal{T} = \left\{ ~g\in {{\mathbb{C}}}[z,t]~\middle\vert \begin{array}{l} \text{There is a monomial } \mathbf{x} \text{ in }\mathcal{B} \text{ such that}\\ \text{the Zariski tangent term of } \sigma^{*}(\mathbf{x}) \text{ is } g. \end{array} \right\}. \]

Let $\mathbf {x} = x^{a}y^{b}z^{c}t^{d}w^{e}$ be a monomial in $\mathcal {L}$. Then $\sigma ^{*}(\mathbf {x})$ is

\[ (z^{2} + f_{{>}2n})^{a} (t^{2} + g_{> 2n})^{b} z^{c}t^{d} = z^{2a + c}t^{2b + d} + h(z,t) \]

where $h(z,\,t)$ is the power series such that the order of $h(z,\,t)$ is greater than $2a+2b+c+d$. Thus the Zariski tangent term of $\sigma ^{*}(\mathbf {x})$ is $z^{2a + c}t^{2b + d}$. It implies that every element of $\mathcal {T}$ is a monomial in ${{\mathbb {C}}}[z,\,t]$.

Lemma 5.2 The number of elements of the set $\mathcal {T}$ is equal to the number of elements of the set $\mathcal {B}$.

Proof. Let $\mathbf {x_1}=x^{a_1}y^{b_1}z^{c_1}t^{d_1}$ and $\mathbf {x_2}=x^{a_2}y^{b_2}z^{c_2}t^{d_2}$ be monomials in the set $\mathcal {B}$ such that the Zariski tangent terms of $\sigma ^{*}(\mathbf {x_1})$ and $\sigma ^{*}(\mathbf {x_2})$ are equal. Then we have

\[ c_1 + 2a_1 = c_2 + 2a_2,\qquad d_1 + 2b_1 = d_2 + 2b_2. \]

Since the two monomials $\mathbf {x_1}$ and $\mathbf {x_2}$ have same degree, we have

\[ a_1 + b_1 + n(c_1 + d_1) = a_2 + b_2 + n(c_2 + d_2). \]

From the above equations, we obtain the equations

\[ a_1 + b_1 = a_2 + b_2,\qquad c_1 + d_1 = c_2 + d_2. \]

If $a_1 = a_2$ then we have $b_1 = b_2$, $c_1 = c_2$ and $d_1 = d_2$. Thus we can assume that $a_1 > a_2$. Then we have $b_1 < b_2$, $c_1 < c_2$ and $d_1 > d_2$. We can write the two monomials $\mathbf {x_1}$ and $\mathbf {x_2}$ as

\[ x^{a_2}y^{b_1}z^{c_1}t^{d_2}x^{a_1 - a_2}t^{d_1 - d_2},\qquad x^{a_2}y^{b_1}z^{c_1}t^{d_2}y^{b_2 - b_1}z^{c_2 - c_1}. \]

They imply that $2(a_1 - a_2) = c_2 - c_1$ and $2(b_2 - b_1) = d_1 - d_2$. We also have $a_1-a_2 = b_2 - b_1$ and $c_2-c_1 = d_1 - d_2$. Thus the two monomials $\mathbf {x_1}$ and $\mathbf {x_2}$ are

\[ x^{a_2}y^{b_1}z^{c_1}t^{d_2}(xt^{2})^{a_1 - a_2},\qquad x^{a_2}y^{b_1}z^{c_1}t^{d_2}(yz^{2})^{a_1 - a_2}. \]

However monomials of the form $(yz^{2})^{\xi } x^{a}y^{b}z^{c}t^{d}$ are not contained in the set $\mathcal {B}$ where $\xi$ is a positive integer. Therefore the two monomials $\mathbf {x_1}$ and $\mathbf {x_2}$ are equal.

By lemma 5.2, we obtain the following.

Corollary 5.3 The set $\mathcal {B}$ is the basis of $\mathcal {L}$.

Proof. We consider the following set:

\[ \mathcal{Z} = \left\{ ~g\in {{\mathbb{C}}}[z,t]~\middle\vert \begin{array}{l} \text{There is a section } s \text{ in }\mathcal{L} \text{ such that}\\ \text{the Zariski tangent term of } \sigma^{*}(s) \text{ is } g. \end{array} \right\}. \]

It is obvious that $\dim _{{{\mathbb {C}}}} \mathcal {Z}\leq \dim _{{{\mathbb {C}}}}\mathcal {L}$. Since $\mathcal {T}\subset \mathcal {Z}$, we have $|\mathcal {T}|\leq \dim _{{{\mathbb {C}}}} \mathcal {Z}$. We also have $\dim _{{{\mathbb {C}}}}\mathcal {L} \leq |\mathcal {B}|$. By lemma 5.2 we have $\dim _{{{\mathbb {C}}}}\mathcal {L} = |\mathcal {B}|$. Consequently, $\mathcal {B}$ is the basis of $\mathcal {L}$.

5.2 Monomial

We consider the ring ${{\mathbb {C}}}[z,\,t]$. The order of monomials in the ring ${{\mathbb {C}}}[z,\,t]$ is the graded lexicographic order with $z< t$. We set $l=h^{0}(S_n,\, \mathcal {O}_{S_n}(k))$. All elements of the basis $\mathcal {B}$ can be written

\[ x^{a_1}y^{b_1}z^{c_1}t^{d_1}w^{e_1},\ldots ,x^{a_l}y^{b_l}z^{c_l}t^{d_l}w^{e_l} \]

in the order of their Zariski tangent terms. we set $a=\sum _{i=1}^{l} a_i$, $b=\sum _{i=1}^{l} b_i$, $c=\sum _{i=1}^{l} c_i$, $d=\sum _{i=1}^{l} d_i$ and $e=\sum _{i=1}^{l} e_i$.

Lemma 5.4 For every basis $\{s_1,\,\ldots s_l\}$ of $\mathcal {L},$ the Newton polygon of the power series by applying the coordinate change $z\mapsto z-\sum _{j>0} \alpha _j t^{j}$ and $t\mapsto t$ to the power series $\prod _{i=1}^{l} \sigma ^{*}(s_i(x,\,y,\,z,\,t,\,1))$ contains the point corresponding to the monomial $z^{c+2a}t^{d+2b}$.

Proof. We set $\xi _i = \sigma ^{*}(x^{a_i}y^{b_i}z^{c_i}t^{d_i}w^{e_i})$ for each $i$. Then the Zariski tangent term of $\xi _i$ is the monomial $z^{c_i+2a_i}t^{d_i+2b_i}$ for each $i$. Let $\zeta _i$ be the power series by applying the coordinate change $z\mapsto z-\sum _{j>0} \alpha _j t^{j}$ and $t\mapsto t$ to $\xi _i$ for each $i$. And let $T$ be the $l\times l$ matrix whose entry in row $i$ and column $j$ is the coefficient of the monomial $z^{c_j+2a_j}t^{d_j + 2b_j}$ of $\zeta _i$. Since the Zariski tangent terms of $\zeta _i$ are $(z-\alpha _1 t)^{c_i+2a_i}t^{d_i + 2b_i}$, all monomials less than $z^{c_i+2a_i}t^{d_i + 2b_i}$ in the monomial ordering are not contained in $\zeta _i$ for each $i$. Thus the matrix $T$ is the upper triangular matrix whose every diagonal entry is $1$.

For any $l\times l$ invertible matrix $M$ there is a permutation matrix $P$ such that $PMT$ is the upper triangular matrix. Then the power series $\eta _i$ with $i=1,\,\ldots l$ given by

\[ \begin{bmatrix} \eta_1\\ \eta_2\\ \vdots\\ \eta_l \end{bmatrix} = PM \begin{bmatrix} \zeta_1\\ \zeta_2\\ \vdots\\ \zeta_l \end{bmatrix} \]

contain the monomial $z^{c_i+2a_i}t^{d_i + 2b_i}$. Thus the Newton polygon of $\prod _{i=1}^{l} \eta _i$ contains the point corresponding to the monomial $z^{c+2a}t^{d+2b}$.

Lemma 5.5 The inequalities $\frac {1}{kl}(c + 2a) \leq \frac {1}{3n} + \frac {2}{3} + \epsilon _k$ and $\frac {1}{kl}(d + 2b) \leq \frac {1}{3n} + \frac {2}{3} + \epsilon _k$ hold where $\epsilon _k$ is a small constant depending on $k$ such that $\epsilon _k \to 0$ as $k\to \infty$.

Proof. We consider the monomials

\[ x^{a_1}y^{b_1}z^{c_1}t^{d_1}w^{e_1},\ldots ,x^{a_l}y^{b_l}z^{c_l}t^{d_l}w^{e_l} \]

of the basis $\mathcal {B}$. Let $B_i$ be the effective Cartier divisor given by $x^{a_i}y^{b_i}z^{c_i}t^{d_i}w^{e_i} = 0$ for each $i$. Then

\[ B {:=} \frac{B_1 + \cdots + B_l}{kl} \]

is the anti-canonical ${{\mathbb {Q}}}$-divisor of $k$-basis type. Moreover $klB$ is given by $x^{a}y^{b}z^{c}t^{d}w^{e} = 0$ where $a=\sum _{i=1}^{l} a_i$, $b=\sum _{i=1}^{l} b_i$, $c=\sum _{i=1}^{l} c_i$, $d=\sum _{i=1}^{l} d_i$ and $e=\sum _{i=1}^{l} e_i$. By corollary 2.8 we have the following inequalities:

\[ \frac{a}{kl}\leq \frac{1}{3} + \epsilon_k,\quad \frac{b}{kl}\leq \frac{1}{3} + \epsilon_k,\quad \frac{c}{kl}\leq \frac{1}{3n} + \epsilon_k,\quad \frac{d}{kl}\leq \frac{1}{3n} + \epsilon_k \]

where $\epsilon _k$ is a small constant depending on $k$ such that $\epsilon _k \to 0$ as $k\to \infty$. Thus we have the inequalities $\frac {1}{kl}(c + 2a) \leq \frac {1}{3n} + \frac {2}{3} + \epsilon _k$ and $\frac {1}{kl}(d + 2b) \leq \frac {1}{3n} + \frac {2}{3} + \epsilon _k$.

5.3 The proof of the theorem 5.1

By using lemmas 4.1 and 5.6 we prove that the log pair $(S_n,\, \lambda D)$ is log canonical, that is, $\delta (S_n) \geq \frac {1}{\lambda } > 1$.

Lemma 5.6 Let $D$ be an anti-canonical ${{\mathbb {Q}}}$-divisor of $k$-basis type on $S_n$ with $k\gg 0$. The log pair $(S_n,\, \lambda D)$ is log canonical at the point $\mathsf p_w$.

Proof. Let $D$ be an anti-canonical ${{\mathbb {Q}}}$-divisor of $k$-basis type on $S_n$ with $k\gg 0$. Then there is a basis $\{s_1,\,\ldots,\,s_l\}$ of the space $H^{0}(S_n,\, \mathcal {O}_{S_n}(k))$ such that

\[ D = \frac{D_1 + \cdots + D_l}{kl} \]

where $D_i$ is the effective divisor of the section $s_i$ for each $i$. In the open set $U$, the effective divisor $\sum _{i=1}^{l} D_i$ is given by the equation $s{:=} \prod _{i=1}^{l} s_i(x,\,y,\,z,\,t,\,1) = 0$. We consider the Newton polygon $N$ of $\sigma ^{*}(s)$ in the coordinates $(u,\,v)$ of ${{\mathbb {R}}}^{2}$. Let $\Lambda$ be the edge of the Newton polygon $N$ that intersects the diagonal line given by $u=v$. If the edge $\Lambda$ is either vertical or horizontal then the log canonical threshold of the log pair $(S_n,\, \sum _{i=1}^{l} D_i)$ at $\mathsf p_w$ is determined by the edge $\Lambda$ (see [14, step A]). By lemma 5.4 the point corresponding to the monomial $z^{c+2a}t^{d+2b}$ is contained in the Newton polygon $N$. Thus we have

\[ \operatorname{lct}_{0}({{\mathbb{C}}}^{2}, (\sigma^{*}(s))\geq \min\left\{\frac{1}{c+2a}, \frac{1}{d+2b}\right\}. \]

By lemma 5.5 we then have

\[ \operatorname{lct}_{0}({{\mathbb{C}}}^{2},\sigma^{*}(s))\geq \frac{\lambda}{kl}. \]

Thus the log pair $(S_n,\, \lambda D)$ is log canonical at the point $\mathsf p_w$.

Suppose that the edge $\Lambda$ is neither vertical nor horizontal. By [14, step C], we can obtain a power series $\eta$ applying a change of coordinates $z\mapsto z-\sum _{j>0} \alpha _j t^{j}$ and $t\mapsto t$ to $\sigma ^{*}(s)$ such that the edge $\Lambda '$ of the Newton polygon $N'$ of the power series $\eta$ that intersects the diagonal line given by $u=v$ determine the log canonical threshold of the log pair $(S_n,\, \sum _{i=1}^{l} D_i)$ at $\mathsf p_w$. By lemma 5.4 the point corresponding to the monomial $z^{c+2a}t^{d+2b}$ is contained in the Newton polygons $N'$ of the power series $\eta$, we have

\[ \operatorname{lct}_{0}({{\mathbb{C}}}^{2}, \eta)\geq \min\left\{\frac{1}{c+2a}, \frac{1}{d+2b}\right\}. \]

By lemma 5.5 we then have

\[ \operatorname{lct}_{0}({{\mathbb{C}}}^{2}, \eta)\geq \frac{\lambda}{kl}. \]

Therefore the log pair $(S_n,\, \lambda D)$ is log canonical at the point $\mathsf p_w$.