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Delta-invariants of complete intersection log del Pezzo surfaces

Published online by Cambridge University Press:  10 May 2022

In-Kyun Kim  and
Joonyeong Won
In-Kyun Kim
Affiliation:
Department of Mathematics, Yonsei University, Seoul, Korea (soulcraw@gmail.com)
Joonyeong Won
Affiliation:
Department of Mathematics, Ewha Womans University, Seoul, Korea (leonwon@ewha.ac.kr)
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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.

Keywords

K-stabilitydel Pezzo surfacecomplete intersectiondelta invariant
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 3 , June 2023 , pp. 1021 - 1036
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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 [Reference Johnson and Kollár9] 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 [Reference Araujo1] 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 [Reference Cheltsov, Park and Shramov2] 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 [Reference Fujita and Odaka8]).

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 [Reference Cheltsov, Rubinstein and Zhang4–Reference Fujita, Liu, Süß, Zhang and Zhuang7, Reference Park and Won14, Reference Park and Won15]. Moreover Li, Tian and Wang generalized in [Reference Li, Tian and Wang13] 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 [Reference Li, Tian and Wang13], the paper [Reference Cheltsov, Park and Shramov3] completes the problem of the existence of the (orbifold) Kähler–Einstein metric on del Pezzo hypersurfaces with $I=1$, case (A):

Theorem 1.2 [Reference Cheltsov, Park and Shramov3]

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 [Reference Kim and Park10] and [Reference Kim and Won11], 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 [Reference Kim and Won11, 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 [Reference Kim and Park10, theorem 1.9] and [Reference Kim and Won11, 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 [Reference Kollár12]

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 [Reference Kollár12]

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 [Reference Fujita and Odaka8]

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 [Reference Fujita and Odaka8]

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 [Reference Fujita and Odaka8, lemma 2.2].

Theorem 2.7 [Reference Cheltsov, Park and Shramov3, 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 [Reference Cheltsov, Park and Shramov3, 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 [Reference Kim and Park10], 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 [Reference Park and Won14, 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 [Reference Park and Won14, 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 $\mathsf p_w$.

Acknowledgments

The authors are very grateful to the referee for valuable suggestions and comments. I.-K. Kim and J. Won were supported by NRF grant funded by the Korea government (MSIT) (I.-K. Kim: NRF-2020R1A2C4002510, J. Won: NRF-2020R1A2C1A01008018). J. Won was supported by the Ewha Womans University Research Grant of 2022.

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