On K-polystability for log del Pezzo pairs of Maeda type

TL;DR

This paper provides an algebraic proof determining the K-polystability of log del Pezzo pairs of Maeda type, extending known results over the complex numbers to a broader algebraic context.

Contribution

It offers a new algebraic proof for the K-polystability of these pairs, complementing previous complex-analytic results.

Findings

Log del Pezzo pairs of Maeda type are K-polystable under certain conditions.

The proof extends known complex results to an algebraic setting.

Clarifies criteria for K-polystability in this class of pairs.

Abstract

We give an algebraic proof for which log del Pezzo pairs of Maeda type are K-polystable or not. If the base field is the complex number field, then the result is already known by Li and Sun.

Equations

$$A_{(X,\Delta )}(F):=1+{\mathrm{coeff}}_F\left(K_{\tilde{X}}-{\pi }^{\ast }(K_X+\Delta )\right).$$

$$K_Y+{\Delta }_Y+\left(1-A_{(X,\Delta )}(F)\right)F={\sigma }^{\ast }(K_X+\Delta ).$$

$$H^0(X,rL-xF):=H^0\left(\tilde{X},{\pi }^{\ast }(rL)\left(\lfloor -xF\rfloor \right)\right)\subset H^0\left(\tilde{X},{\pi }^{\ast }(rL)\right)$$

$$\mathrm{vol}(L-xF):=\underset{r\to \infty }{\mathrm{limsup}}\frac{{\dim }_{\mathbb{k}}{H}^0(X,rL-rxF)}{r^n/n!}.$$

$$\tau (F):=\sup \{x\in {\mathbb{R}}_{\ge 0}|\mathrm{vol}(L-xF)>0\}.$$

$$\epsilon (F):=\max \{x\in {\mathbb{R}}_{\ge 0}|{\sigma }^{\ast }L-xF\text{ is nef on }Y\},$$

$${\hat{\beta }}_{(X,\Delta )}(F):=1-\frac{{\int }_0^{\infty }\mathrm{vol}(L-xF)dx}{A_{(X,\Delta )}(F)\cdot (L^{\cdot n})}.$$

$$\underset{k,j\in {\mathbb{Z}}_{\ge 0}}{⨁}{H}^0(X,krL-jF)$$

$$\mathbb{k}(X{)}^{\ast }\stackrel{{\rho }^{\ast }}{\to }\mathbb{k}(X)(t{)}^{\ast }\stackrel{{\mathrm{ord}}_{(t^{-1})}}{\to }\mathbb{Z},$$

$$\mathrm{Aut}(X,\Delta ):=\{\theta \in \mathrm{Aut}(X)|{\theta }_{\ast }\Delta =\Delta \}\subset \mathrm{Aut}(X),$$

$${\hat{\beta }}_{(X,\Delta )}(F)\ge \frac{1}{n+1}.$$

$$f(y)\le f(x){\left(\frac{y-x}{n}\frac{f^{\prime }(x)}{f(x)}+1\right)}^n.$$

$$\tau (F)\le x+\frac{nf(x)}{-f^{\prime }(x)}.$$

$$f(x{)}^{1/n}\ge \frac{y-x}{y-c}f(c{)}^{1/n}+\frac{x-c}{y-c}f(y{)}^{1/n}$$

$$\frac{1}{y-x}f(y{)}^{1/n}$$

$$Y\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$f_0:=-\frac{1}{(F^{\cdot 2}{)}_Y}.$$

$${\hat{\beta }}_{(X,\Delta )}(F)=1-\frac{\epsilon (F)+\tau (F)}{3\cdot A_{(X,\Delta )}(F)}.$$

$$\mathrm{vol}(L-xF)=(L^{\cdot 2})-\frac{x^2}{f_0}$$

$$(L^{\cdot 2})-\frac{\epsilon (F{)}^2}{f_0}$$

$$-\frac{2\epsilon (F)}{f_0}$$

$$\pi :\tilde{X}=X_m\to \cdots \to X_1\to X_0=X$$

$$E_i^{\ast }:=E_{q(i)}^{\ast }+E_{i-1}^{\ast }+({\pi }_{i+1}\circ \cdots \circ {\pi }_m{)}^{\ast }{E}_i.$$

$$K_{\tilde{X}}+{\pi }_{\ast }^{-1}\Delta -{\pi }^{\ast }(K_X+\Delta )\le \frac{A_{(X,\Delta )}(F)-1}{f}{E}_m^{\ast }.$$

$$H^0\left(\tilde{X},k\left({\pi }^{\ast }L-\frac{j}{f}{E}_m^{\ast }\right)\right)\to H^0\left(\tilde{X},k\left({\pi }^{\ast }L-jF\right)\right)$$

$${\nu }^{\ast }F=\frac{1}{f}{E}_m^{\ast }.$$

$${\pi }^{\ast }P=:\tilde{P}+\sum _{i=1}^m{n}_i{\tilde{E}}_i,$$

$$M:={\pi }^{\ast }P-\frac{x}{f}{E}_m^{\ast }={\nu }^{\ast }\left({\sigma }^{\ast }P-xF\right).$$

$$\left(E_i^{\ast }\cdot {\tilde{E}}_j\right)=\left(\left(E_{q(i)}^{\ast }+E_{i-1}^{\ast }+({\pi }_{i+1}\circ \cdots \circ {\pi }_m{)}^{\ast }{E}_i\right)\cdot {\tilde{E}}_j\right).$$

$$\left(E_i^{\ast }\cdot {\tilde{E}}_i\right)=\left(({\pi }_{i+1}\circ \cdots \circ {\pi }_m{)}^{\ast }{E}_i\cdot {\tilde{E}}_i\right)=-1.$$

$$\left(({\pi }_{i+1}\circ \cdots \circ {\pi }_m{)}^{\ast }{E}_i\cdot {\tilde{E}}_j\right)=1,$$

Full text

1. 1 Introduction
2. 2 K-stability of log Fano pairs
3. 3 Basic properties of surfaces
4. 4 Product-type prime divisors
5. 5 On the projective plane
6. 6 On the product of the projective lines
7. 7 Proof of Corollary 1.2

1. Introduction

We work over an arbitrary algebraically closed field $\Bbbk$ with the characteristic zero. Let $X$ be a Fano manifold, that is, $X$ is a smooth projective variety over $\Bbbk$ such that the anti-canonical divisor $-K_{X}$ is ample. We are interested in the problem whether $X$ is K-polystable or not. In fact, if $\Bbbk$ is equal to the complex number field $\mathbb{C}$, then K-polystability of $X$ is known to be equivalent to the existence of Kähler-Einstein metrics on $X$ thanks to the works [Tia97, Don02, Sto09, Ber16, CDS15a, CDS15b, CDS15c, Tia15] and references therein. It is natural to consider K-polystability for not only Fano manifolds but also log Fano pairs $(X,\Delta)$ (see Definition 2.1 (4)). However, in general, it is difficult to test K-polystability purely algebraically. Recently, Li, Wang and Xu in [LWX18, Theorem 1.4] gave a purely algebraic proof for which toric log Fano pairs are K-polystable or not. However, when a log Fano pair is not a toric pair, it is difficult to test K-polystability. See also Remark 6.2.

In this article, we mainly consider K-polystability of log del Pezzo pairs, that is, log Fano pairs of dimension two. The purpose of this article is to give an algebraic proof for K-polystability of the log del Pezzo pair $(\mathbb{P}^{2},\delta C)$, where $\delta$ is a non-negative rational number with $\delta<3/4$ and $C\subset\mathbb{P}^{2}$ is a smooth conic, and the log del Pezzo pair $(\mathbb{P}^{1}\times\mathbb{P}^{1},\delta C)$, where $\delta$ is a non-negative rational number with $\delta<1/2$ and $C\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ is the diagonal.

Theorem 1.1** (cf. [LS14, Example 3.12]).**

1. (1)

*Assume that $C\subset\mathbb{P}^{2}$ is a smooth conic and let $\delta\in[0,1)\cap\mathbb{Q}$. Then the log del Pezzo pair $(\mathbb{P}^{2},\delta C)$ is K-polystable $($resp., K-semistable$)$ if and only if $\delta<3/4$ *$($resp., \delta\leq 3/4$$). 2. (2)

*Assume that $C\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ is the diagonal and let $\delta\in(0,1)\cap\mathbb{Q}$. Then the log del Pezzo pair $(\mathbb{P}^{1}\times\mathbb{P}^{1},\delta C)$ is K-polystable $($resp., K-semistable$)$ if and only if $\delta<1/2$ *$($resp., \delta\leq 1/2$$).

If $\Bbbk=\mathbb{C}$, then the above result is known by [LS14, Example 3.12] and [Ber16, Theorem 4.8]. We emphasize that, our proof is based on the work [Fuj17a], purely algebraic, direct and easy. Moreover, in Theorem 1.1 (1), we give a very easy and purely algebraic proof for K-polystability of $\mathbb{P}^{2}$. For the proof of Theorem 1.1 (2), we use the fact $\mathbb{P}^{1}\times\mathbb{P}^{1}$ is K-semistable. We can prove this fact purely algebraically (see [Kem78, Li17, Blu16, BJ17]).

As an immediate consequence of Theorem 1.1, we get an algebraic proof for the classification of K-polystable log del Pezzo pairs of Maeda type. A pair $(X,\Delta)$ is said to be a log del Pezzo pair of Maeda type if $X$ is a smooth projective surface and $\Delta$ is a nonzero effective $\mathbb{Q}$-divisor on $X$ such that $D:=\operatorname{Supp}\Delta$ is simple normal crossing and both $-(K_{X}+\Delta)$ and $-(K_{X}+D)$ are ample.

Corollary 1.2**.**

Let $(X,\Delta)$ be a log del Pezzo pair of Maeda type. Then $(X,\Delta)$ is K-polystable $($resp., K-semistable$)$ if and only if

• •

$(X,\Delta)$* is isomorphic to $(\mathbb{P}^{2},\delta C)$ with $C$ a smooth conic and $\delta<3/4$ *$($resp., \delta\leq 3/4$$), or

• •

$(X,\Delta)$* is isomorphic to $(\mathbb{P}^{1}\times\mathbb{P}^{1},\delta C)$ with $C$ the diagonal and $\delta<1/2$ *$($resp., \delta\leq 1/2$$).

For the proof, we use Maeda’s classification result [Mae86]. In general, Cheltsov and Rubinstein gave a question in [CR15] that which asymptotically log del Pezzo pairs (see [CR15, Definition 1.1]) are K-polystable or not. In order to consider the question, it is important to establish techniques to test K-polystability of log del Pezzo pairs. The above log del Pezzo pairs $(X,\Delta)$ in Theorem 1.1 are no longer uniformly K-stable (see Theorems 5.1 and 6.3). Hence we cannot apply the techniques to evaluate the delta invariants introduced in [FO16, BJ17] in order to show K-polystability. The proof of Theorem 1.1 will be important to answer the question of Cheltsov and Rubinstein.

This article is organized as follows. In §2, we give the definitions for K-polystability and K-semistability of log Fano pairs. The definitions are not of original form in [Tia97, Don02]. Moreover, we see several numerical properties of the invariants $\hat{\beta}_{(X,\Delta)}(F)$ for log del Pezzo pairs $(X,\Delta)$ and of dreamy prime divisors $F$ over $(X,\Delta)$. In §3, we see basic properties for exceptional prime divisors over smooth surfaces. Moreover, we see that there exists a non-dreamy prime divisor over $\mathbb{P}^{2}$. In §4, we discuss product-type prime divisors over $(\mathbb{P}^{2},\delta C)$ and $(\mathbb{P}^{1}\times\mathbb{P}^{1},\delta C)$. In §5, we prove Theorem 1.1 (1); in §6, we prove Theorem 1.1 (2); in §7, we prove Corollary 1.2.

Acknowledgments**.**

This work was supported by JSPS KAKENHI Grant Number 18K13388.

For the minimal model program, we refer the readers to [KM98]. For a birational map $X\dashrightarrow X^{\prime}$ between normal projective varieties and for a $\mathbb{Q}$-divisor $\Delta$ on $X$, the strict transform of $\Delta$ on $X^{\prime}$ is denoted by $\Delta^{X^{\prime}}$. Moreover, for a prime divisor $E$ on $X$, the coefficient of $\Delta$ at $E$ is denoted by $\operatorname{coeff}_{E}\Delta$.

For the toric geometry, we refer the readers to [CLS11]. In this article, we consider only $2$-dimensional toric varieties. We always fix the lattice $N:=\mathbb{Z}^{\oplus 2}$ of rank $2$ and set $N_{\mathbb{R}}:=N\otimes_{\mathbb{Z}}\mathbb{R}$.

2. K-stability of log Fano pairs

We recall K-polystability and K-semistability of log Fano pairs in [Fuj17a, Fuj17b]. The definition is equivalent to the original one [Tia97, Don02] by the works [Li17, Fuj16, Fuj17a, Fuj17b].

Definition 2.1**.**

Let $(X,\Delta)$ be a log pair, that is, $X$ is a normal variety and $\Delta$ is an effective $\mathbb{Q}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier. Let $F$ be a prime divisor over $X$, that is, there exists a resolution $\pi\colon\tilde{X}\to X$ such that $F$ is a prime divisor on $\tilde{X}$.

1. (1)

We set

[TABLE]

The center (i.e., the image) of $F$ on $X$ is denoted by $c_{X}(F)$. We recall that the pair $(X,\Delta)$ is said to be klt if $A_{(X,\Delta)}(F)>0$ for any prime divisor $F$ over $X$. 2. (2)

([Ish04]) The $F$ is said to be primitive over $X$ if there exists a projective birational morphism $\sigma\colon Y\to X$ (called the extraction of $F$) with $Y$ normal such that $-F$ is a $\sigma$-ample $\mathbb{Q}$-Cartier divisor on $Y$. 3. (3)

([Sho96, Pro00]) The $F$ is said to be plt-type over $(X,\Delta)$ if $F$ is primitive over $X$ and $(Y,\Delta_{Y}+F)$ is plt, where $\sigma\colon Y\to X$ is the extraction of $F$ and $\Delta_{Y}$ is the $\mathbb{Q}$-divisor on $Y$ given by the equation

[TABLE] 4. (4)

The pair $(X,\Delta)$ is said to be a log Fano pair if $(X,\Delta)$ is a projective klt pair such that $-(K_{X}+\Delta)$ is an ample $\mathbb{Q}$-divisor on $X$. If moreover the dimension of $X$ is equal to $2$, then we call it a log del Pezzo pair.

Definition 2.2** (see [Li17, Fuj16, Fuj17a, Fuj17b]).**

Let $(X,\Delta)$ be an $n$-dimensional log Fano pair and set $L:=-(K_{X}+\Delta)$. Take any prime divisor $F$ over $X$ and let us fix a resolution $\pi\colon\tilde{X}\to X$ such that $F$ is a prime divisor on $\tilde{X}$.

1. (1)

For any $x\in\mathbb{R}_{\geq 0}$ and for any $r\in\mathbb{Z}_{\geq 0}$ with $rL$ Cartier, let $H^{0}(X,rL-xF)$ be the subspace of $H^{0}(X,rL)$ given by

[TABLE]

under the natural identity $H^{0}(X,rL)=H^{0}\left(\tilde{X},\pi^{*}(rL)\right)$. 2. (2)

For any $x\in\mathbb{R}_{\geq 0}$, we set

[TABLE]

We set

[TABLE]

Moreover, if $F$ is primitive over $X$, then we set

[TABLE]

where $\sigma\colon Y\to X$ is the extraction of $F$. Obviously, we have $\varepsilon(F)\leq\tau(F)$. 3. (3)

We set

[TABLE] 4. (4)

The $F$ is said to be dreamy over $(X,\Delta)$ if the graded $\Bbbk$-algebra

[TABLE]

is finitely generated over $\Bbbk$ for some $r\in\mathbb{Z}_{>0}$ with $rL$ Cartier. 5. (5)

The $F$ is said to be product-type over $(X,\Delta)$ if there exists a $1$-parameter subgroup $\rho\colon\mathbb{G}_{m}\to\operatorname{Aut}(X,\Delta)$ of $\operatorname{Aut}(X,\Delta)$ such that the divisorial valuation $\operatorname{ord}_{F}\colon\Bbbk(X)^{*}\to\mathbb{Z}$ is equal to the composition

[TABLE]

where

[TABLE]

and $\rho^{*}\colon\Bbbk(X)\to\Bbbk(X)(t)$ is given by the natural morphism $\rho\colon\mathbb{G}_{m}\times X\to X$.

Remark 2.3**.**

1. (1)

The above definitions are not depend on the choice of the morphism $\pi\colon\tilde{X}\to X$. 2. (2)

The function $\operatorname{vol}(L-xF)$ is continuous and non-increasing over $x\in[0,\infty)$ by [Laz04a, Laz04b]. Moreover, by [BFJ09, Theorem A], $\operatorname{vol}(L-xF)$ is $\mathcal{C}^{1}$ over $x\in[0,\tau(F))$. 3. (3)

By [Ish04, Proposition 2.4], the extraction of $F$ is unique if exists. 4. (4)

If $F$ is product-type over $(X,\Delta)$, then $F$ is dreamy over $(X,\Delta)$ by [Fuj17b, Proposition 3.10]. If $F$ is dreamy over $(X,\Delta)$, then $F$ is primitive over $X$ by [Fuj17a, Remark 1.3 (1)].

Definition 2.4**.**

Let $(X,\Delta)$ be a log Fano pair.

1. (1)

The pair $(X,\Delta)$ is said to be K-semistable (resp., K-stable) if $\hat{\beta}_{(X,\Delta)}(F)\geq 0$ (resp., $>0$) for any dreamy prime divisor $F$ over $(X,\Delta)$. 2. (2)

The pair $(X,\Delta)$ is said to be K-polystable if K-semistable, and a dreamy prime divisor $F$ over $(X,\Delta)$ satisfies that $\hat{\beta}_{(X,\Delta)}(F)=0$ only if $F$ is a product-type over $(X,\Delta)$. 3. (3)

The pair $(X,\Delta)$ is said to be uniformly K-stable if there exists $\varepsilon>0$ such that $\hat{\beta}_{(X,\Delta)}(F)\geq\varepsilon$ for any dreamy prime divisor $F$ over $(X,\Delta)$.

Remark 2.5**.**

1. (1)

By the works [Li17, Fuj16, Fuj17a, Fuj17b], the notions of K-semistability, K-polystability, K-stability and uniform K-stability are equivalent to the original one in [Tia97, Don02, LX14]. 2. (2)

It is known that K-semistability (resp., uniform K-stability) of $(X,\Delta)$ is equivalent to the condition $\hat{\beta}_{(X,\Delta)}(F)\geq 0$ (resp., $\hat{\beta}_{(X,\Delta)}(F)\geq\varepsilon$) for any prime divisor $F$ over $X$. See [Li17, Fuj16, Fuj17a].

We recall the following:

Proposition 2.6**.**

Let $(X,\Delta)$ be an $n$-dimensional log Fano pair and let $F$ be a prime divisor over $X$. If $\tau(F)\leq A_{(X,\Delta)}(F)$, then we have the inequality

[TABLE]

Proof.

Follows immediately from [Fuj17a, Proposition 2.1]. ∎

The following proposition is essential in §6.

Proposition 2.7**.**

Let $(X,\Delta)$ be an $n$-dimensional log Fano pair, let $L:=-(K_{X}+\Delta)$, and let $F$ be a prime divisor over $X$. Set $f(x):=\operatorname{vol}(L-xF)$ for $x\in\mathbb{R}_{\geq 0}$. Then, for any $0\leq x<y\leq\tau(F)$, we have the following inequality

[TABLE]

In particular, if $x>0$, then we have

[TABLE]

Proof.

We may assume that $x>0$. For any $c\in[0$, $x)$, we have

[TABLE]

by the log-concavity of the volume functions (see, e.g., [LM09]). Thus we have

[TABLE]

When $0<x<\tau(F)$, we know that $f^{\prime}(x)<0$ (see [LM09, Corollary 4.27] for example). Hence we get the assertion. ∎

From now on, let us assume that $(X,\Delta)$ is a log del Pezzo pair with $\rho(X)=1$, where $\rho(X)$ is the Picard number of $X$. By [KM98, Proposition 4.11], $X$ is $\mathbb{Q}$-factorial. Take any dreamy exceptional prime divisor $F$ over $(X,\Delta)$. By Remark 2.3 (4), $F$ is primitive over $X$. Let $\sigma\colon Y\to X$ be the extraction of $F$. Then $Y$ is $\mathbb{Q}$-factorial by [Pro00, Remark 2.2 (i)]. Moreover, by [KKL16, Theorem 4.2], we have $\varepsilon(F)$, $\tau(F)\in\mathbb{Q}_{>0}$ and $\sigma^{*}L-\varepsilon(F)F\,\,(\not\sim_{\mathbb{Q}}0)$ induces a non-trivial morphism $\mu\colon Y\to Z$ with connected fibers and with $Z$ normal and $\rho(Z)=1$. If $\varepsilon(F)<\tau(F)$, then $\mu$ is birational; if $\varepsilon(F)=\tau(F)$, then $Z\simeq\mathbb{P}^{1}$.

Definition 2.8**.**

The above diagram

[TABLE]

is called the standard diagram with respects to $F$.

We frequently use the following lemma:

Lemma 2.9**.**

Let $G\subset Y$ be an irreducible curve.

1. (1)

If $\mu_{*}G=0$, then we have $\left(\left(\sigma^{*}L-\varepsilon(F)F\right)\cdot G\right)=0$. 2. (2)

If $\mu_{*}G\neq 0$, then we have $\left(\left(\sigma^{*}L-\tau(F)F\right)\cdot G\right)\geq 0$.

Proof.

(1) The assertion is obvious since the $\mathbb{Q}$-divisor $\sigma^{*}L-\varepsilon(F)F$ is the pullback of a $\mathbb{Q}$-divisor on $Z$.

(2) If $\varepsilon(F)=\tau(F)$, then the assertion is trivial since $\sigma^{*}L-\varepsilon(F)F$ is nef. If $\varepsilon(F)<\tau(F)$, then $\sigma^{*}L-\tau(F)F$ is $\mathbb{Q}$-linearly equivalent to some positive multiple of the $\mu$-exceptional curve. Thus the assertion follows. ∎

The following lemma is proved as in the case with the proof of [Fuj17a, Claim 4.3].

Lemma 2.10**.**

Let us set $L:=-(K_{X}+\Delta)$ and

[TABLE]

Then we have $\varepsilon(F)\cdot\tau(F)=f_{0}\cdot(L^{\cdot 2})$ and

[TABLE]

Proof.

We recall the proof of [Fuj17a, Claim 4.3]. Note that

[TABLE]

for any $x\in[0,\varepsilon(F)]$. If $\varepsilon(F)=\tau(F)$, then the assertion is trivial since we know that $\operatorname{vol}(L-\tau(F)F)=0$. We may assume that $\varepsilon(F)<\tau(F)$. Then, since $\mu_{*}\sigma^{*}L$ and $\mu_{*}F$ are $\mathbb{Q}$-linearly proportional on $Z$, there exists $c_{0}\in\mathbb{Q}_{>0}$ such that, for any $x\in[\varepsilon(F),\tau(F)]$, we have $\operatorname{vol}(L-xF)=c_{0}(\tau(F)-x)^{2}$. When we apply Remark 2.3 (2) with $x=\varepsilon(F)$, we have

[TABLE]

Thus the assertion follows. ∎

3. Basic properties of surfaces

In this section, we see basic properties for exceptional prime divisors on surfaces in order to prove Theorem 1.1.

3.1. Sequences of monoidal transforms

Definition 3.1**.**

Let $X$ be a smooth surface and let $F$ be an exceptional prime divisor over $X$. We construct the sequence

[TABLE]

of monoidal transform (called the sequence of monoidal transforms with respects to $F$) given by:

1. (1)

$X_{0}:=X$. 2. (2)

If $F$ is a prime divisor on $X_{i}$, then we set $m:=i$, $\tilde{X}:=X_{m}$ and we stop the construction. 3. (3)

If $F$ is exceptional over $X_{i}$, then we set $p_{i+1}:=c_{X_{i}}(F)$, let $\pi_{i+1}\colon X_{i+1}\to X_{i}$ be the blowup along $p_{i+1}$ and let $E_{i+1}\subset X_{i+1}$ be the $\pi_{i+1}$-exceptional curve.

For any $1\leq i\leq m$, let $\tilde{E}_{i}\subset\tilde{X}$ be the strict transform of $E_{i}$ on $\tilde{X}$. Obviously, we have $p_{i+1}\in E_{i}$ for any $1\leq i\leq m-1$, and we have $\tilde{E}_{m}=F$.

Definition 3.2**.**

Under the notation in Definition 3.1, we define the following notions:

1. (1)

For any $2\leq i\leq m$, let us define $q(i)\in[0,i-2]\cap\mathbb{Z}$ as follows:

• •

If $p_{i}\not\in E_{j}^{X_{i-1}}$ for any $1\leq j\leq i-2$, then we set $q(i):=0$.

• •

If $p_{i}\in E_{j}^{X_{i-1}}$ for some $1\leq j\leq i-2$, then we set $q(i):=j$.

Since $E_{1}^{X_{i-1}},\dots,E_{i-2}^{X_{i-1}}$, $E_{i-1}$ are simple normal crossing and $p_{i}\in E_{i-1}$, the definition makes sense. 2. (2)

For any $0\leq i\leq m$, let us define the effective $\mathbb{Z}$-divisor $E_{i}^{*}$ on $\tilde{X}$ as follows:

• •

We set $E_{0}^{*}:=0$ and $E_{1}^{*}:=(\pi_{2}\circ\cdots\circ\pi_{m})^{*}E_{1}$.

• •

For any $2\leq i\leq m$, we set

[TABLE] 3. (3)

We set $f:=\operatorname{coeff}_{F}E_{m}^{*}\in\mathbb{Z}_{>0}$.

Lemma 3.3**.**

1. (1)

For any $1\leq i$, $j\leq m$, we have $\left(E_{i}^{*}\cdot\tilde{E}_{j}\right)=-\delta_{ij}$. 2. (2)

For any effective $\mathbb{Q}$-divisor $\Delta$ on $X$, we have

[TABLE] 3. (3)

For any $L\in\operatorname{Pic}X$, $j\in\mathbb{Q}_{\geq 0}$ and for any $k\in\mathbb{Z}_{>0}$ with $k/f$, $kj/f\in\mathbb{Z}$, the natural homomorphism

[TABLE]

given by the effective divisor $kj((1/f)E_{m}^{*}-F)$ is an isomorphism. 4. (4)

Assume that $F$ is primitive over $X$ and let $\sigma\colon Y\to X$ be the extraction of $F$. Then the natural morphism $\nu\colon\tilde{X}\to Y$ over $X$ is the minimal resolution of $Y$, and we have the equality

[TABLE] 5. (5)

Assume furthermore that $X$ is projective. Take an effective and nef $\mathbb{Z}$-divisor $P$ on $X$. Set

[TABLE]

where $\tilde{P}:=\pi_{*}^{-1}P$. Take any $x\in[0,n_{m}]$ and set

[TABLE]

If each irreducible component $P^{\prime}$ of $\tilde{P}$ is nef, or more generally $(M\cdot P^{\prime})\geq 0$, then $M$ is nef.

Proof.

(1) Let us remark that $E_{i}^{*}=(\pi_{i+1}\circ\dots\circ\pi_{m})^{*}(E_{i}^{*})^{X_{i}}$ for any $1\leq i\leq m$. Thus it is obvious that $\left(E_{1}^{*}\cdot\tilde{E}_{j}\right)=-\delta_{1j}$. From now on, let us assume that $i\geq 2$. We may assume that $\left(E_{i^{\prime}}^{*}\cdot\tilde{E}_{j}\right)=-\delta_{i^{\prime}j}$ holds for any $1\leq i^{\prime}<i$ and for any $1\leq j\leq m$. From the construction, we have the equality

[TABLE]

If $j>i$, then we have $\left(E_{i}^{*}\cdot\tilde{E}_{j}\right)=0$ since $\tilde{E}_{j}$ is exceptional over $X_{i}$. If $j=i$, then we have

[TABLE]

If $j\in\{q(i)$, $i-1\}$, since $\left(\left(E_{q(i)}^{*}+E_{i-1}^{*}\right)\cdot\tilde{E}_{j}\right)=-1$ and

[TABLE]

we have $\left(E_{i}^{*}\cdot\tilde{E}_{j}\right)=0$. If $j<i$ and $j\not\in\{q(i)$, $i-1\}$, then we have

[TABLE]

Thus we have $\left(E_{i}^{*}\cdot\tilde{E}_{j}\right)=0$.

(2) For any $1\leq i\leq m-1$, the self intersection number of $\tilde{E}_{i}$ is smaller than or equal to $-2$. In particular, we have $\left(K_{\tilde{X}}\cdot\tilde{E}_{i}\right)\geq 0$. Set

[TABLE]

By (1), we have

[TABLE]

for any $1\leq j\leq m-1$. Hence we have $h_{i}\leq 0$ for any $1\leq i\leq m-1$ by [KM98, Lemma 3.41].

(3) Take any effective divisor $G$ on $X$ and set $g:=\operatorname{coeff}_{F}\pi^{*}G\in\mathbb{Z}_{\geq 0}$. Set

[TABLE]

By (1), we have

[TABLE]

for any $1\leq j\leq m-1$. Again by [KM98, Lemma 3.41], we have $g_{i}\leq 0$ for any $1\leq i\leq m-1$. Thus we have $\pi^{*}G\geq(g/f)E_{m}^{*}$. The assertion follows from this fact.

(4) The set of $\nu$-exceptional curves is equal to $\{\tilde{E}_{i}\}_{1\leq i\leq m-1}$. Moreover, we have $\left(K_{\tilde{X}}\cdot\tilde{E}_{i}\right)\geq 0$ for any $1\leq i\leq m-1$. Thus $\nu$ is the minimal resolution of $Y$. Since $\left(E_{m}^{*}\cdot\tilde{E}_{i}\right)=0$ for any $1\leq i\leq m-1$, we have the equality $\nu^{*}F=(1/f)E_{m}^{*}$.

(5) Since $\nu_{*}M=\nu_{*}\left(\tilde{P}+(n_{m}-x)\tilde{E}_{m}\right)$, the $\mathbb{R}$-divisor

[TABLE]

is effective. Thus, if $P^{\prime}$ is nef, then we get $(M\cdot P^{\prime})\geq 0$. Obviously, we have $\left(M\cdot\tilde{E}_{m}\right)=x/f\geq 0$. From the assumption, for any irreducible curve $B$ on $Y$, we have $\left(M\cdot\nu_{*}^{-1}B\right)\geq 0$. Moreover, for any $\nu$-exceptional curve $G$ on $\tilde{X}$, we have $(M\cdot G)=0$. ∎

Definition 3.4**.**

Under the notations in Definitions 3.1 and 3.2, let us further assume that $F$ is plt-type over $X$. From the construction, $F$ on $\tilde{X}$ intersects $\tilde{E}_{1}\cup\cdots\cup\tilde{E}_{m-1}$ at most $2$ points. Moreover, by [KM98, Theorem 4.15], the dual graph of $\tilde{E}_{1}\cup\dots\cup\tilde{E}_{m}$ is a straight chain.

1. (1)

Set $k:=\max\{2\leq i\leq m\,\,|\,\,q(i)=0\}$. From the structure of the dual graph of $\tilde{E}_{1}\cup\dots\cup\tilde{E}_{m}$, we have $q(i)=0$ for any $2\leq i\leq k$. 2. (2)

For any $0\leq i\leq m$, let us define $a_{i}$, $b_{i}\in\mathbb{Z}_{\geq 0}$ as follows:

• •

$(a_{0}$, $b_{0}):=(1$, $0)$, $(a_{1}$, $b_{1}):=(1$, $1)$.

• •

$(a_{i}$, $b_{i}):=(a_{q(i)}$, $b_{q(i)})+(a_{i-1}$, $b_{i-1})$.

Moreover, let us set $a^{F}:=a_{m}$ and $b^{F}:=b_{m}$. Clearly, we have $a_{i}\geq b_{i}$ for any $0\leq i\leq m$.

Lemma 3.5**.**

1. (1)

For any $1\leq i\leq m$, $a_{i}$ and $b_{i}$ are mutually prime. In particular, $a^{F}$ and $b^{F}$ are mutually prime. 2. (2)

For any $1\leq i\leq m$, we have $A_{X}(E_{i})=a_{i}+b_{i}$. In particular, we have $A_{X}(F)=a^{F}+b^{F}$. 3. (3)

We have $f=a^{F}b^{F}$ and $k=\lceil(a^{F}/b^{F})\rceil$, where $f$ be as in Definition 3.1. 4. (4)

For any $1\leq i\leq k$, we have $\operatorname{coeff}_{\tilde{E}_{i}}E_{m}^{*}=\min\{ib^{F},\,a^{F}\}$.

Proof.

All of the assertions are étale local. By [Pro01, Proposition 6.2.6], we may assume that $X=\mathbb{A}^{2}$ and $\sigma\colon Y\to X$ is a toric morphism of toric varieties. Thus, there exist mutually prime $a$, $b\in\mathbb{Z}_{>0}$ with $a\geq b$ such that:

• •

$X$ corresponds to the fan $\Sigma_{0}$ in $N_{\mathbb{R}}$ (i.e., $X=X_{\Sigma_{0}}$) such that $\Sigma_{0}$ consists of the $2$-dimensional cone $\mathbb{R}_{\geq 0}(1,0)+\mathbb{R}_{\geq 0}(0,1)$ and all of its faces.

• •

$Y$ corresponds to the fan $\Sigma^{\prime}$ in $N_{\mathbb{R}}$ (i.e., $Y=X_{\Sigma^{\prime}}$) such that $\Sigma^{\prime}$ consists of the $2$-dimensional cones $\mathbb{R}_{\geq 0}(1,0)+\mathbb{R}_{\geq 0}(a,b)$, $\mathbb{R}_{\geq 0}(a,b)+\mathbb{R}_{\geq 0}(0,1)$, and all of those faces.

• •

The morphism $\sigma\colon Y\to X$ corresponds to the natural morphism of fans.

Let us consider the sequence of monoidal transforms with respects to $F$. Every step of the monoidal transform is a toric morphism. Let $\Sigma_{i}$ in $N_{\mathbb{R}}$ be the fan associates with $X_{i}$. Assume that $i<m$. Then $(a$, $b)\in N_{\mathbb{R}}$ belongs to the interior of some $2$-dimensional cone $\sigma_{i}\in\Sigma_{i}$. Moreover, $p_{i+1}$ is the torus-invariant point in $X_{i+1}$ corresponds to $\sigma_{i}$ and $\Sigma_{i+1}$ is obtained by the star subdivision of $\Sigma_{i}$ along $\sigma_{i}$ (see [CLS11, Definition 3.3.13]).

Let $(a^{i}$, $b^{i})\in N$ be the primitive generator of the $1$-dimensional cone corresponds to $E_{i}\subset X_{i}$. Since $\sigma_{i}=\mathbb{R}_{\geq 0}(1,0)+\mathbb{R}_{\geq 0}(i,1)$ for any $0\leq i<a/b$, we have $(a^{i}$, $b^{i})=(i$, $1)$ for any $1\leq i<a/b+1$. Therefore, we have $q(i)=0$ for any $1\leq i\leq\lceil(a/b)\rceil$. Moreover, since

[TABLE]

we have $q(i)\neq 0$ for any $i>\lceil(a/b)\rceil$. Thus we have $k=\lceil(a/b)\rceil$. Moreover, from the construction, we have the equality

[TABLE]

for any $2\leq i\leq m$, where we set $(a^{0}$, $b^{0}):=(1$, $0)$. Hence we can inductively show that $(a_{i}$, $b_{i})=(a^{i}$, $b^{i})$ for any $1\leq i\leq m$. In particular, we have $(a^{F}$, $b^{F})=(a$, $b)$.

(1) The assertion is trivial since $a^{i}$ and $b^{i}$ are mutually prime.

(2) We know that

[TABLE]

Thus the assertion follows inductively.

(3) We have already seen that $k=\lceil(a/b)\rceil=\lceil(a^{F}/b^{F})\rceil$. Let $l_{1}$, $l_{2}\subset Y$ be the torus invariant curve on $Y$ corresponds to the $1$-dimensional cone $\mathbb{R}_{\geq 0}(1,0)$, $\mathbb{R}_{\geq 0}(0,1)$, respectively. Then we have $(F^{\cdot 2})_{Y}=-1/(ab)$ since $(F\cdot l_{1})_{Y}=1/b$ and $(F\cdot l_{2})_{Y}=1/a$ (see [CLS11, Theorem 15.1.1]). Thus we have

[TABLE]

by Lemma 3.3.

(4) The dual graph of $\tilde{E}_{1}\cup\cdots\cup\tilde{E}_{m}$ on $\tilde{X}$ is of the form:

\tilde{E}_{1}$$\tilde{E}_{k-1}the other components$\tilde{E}_{k}$

Moreover, $l_{1}^{\tilde{X}}$ (resp., $l_{2}^{\tilde{X}}$) intersects $\tilde{E}_{k}$ (resp., $\tilde{E}_{1}$) transversally. Thus we get

[TABLE]

Thus the assertion is true when $i=1$ or $k$. We note that

[TABLE]

for $3\leq i\leq k-1$. Thus the assertion follows inductively. ∎

3.2. Dreamy prime divisors over log del Pezzo pairs

We see basic properties of primes divisors over log del Pezzo pairs.

Proposition 3.6**.**

Let $(X,\Delta)$ be a log del Pezzo pair and let $F$ be a prime divisor over $X$ with $\hat{\beta}_{(X,\Delta)}(F)<1/3$. Then $F$ is dreamy over $(X,\Delta)$.

Proof.

Set $L:=-(K_{X}+\Delta)$. By Proposition 2.6, we have $\tau(F)>A_{(X,\Delta)}(F)$. Let $\phi\colon X_{0}\to X$ be the minimal resolution of $X$ and set $K_{X_{0}}+\Delta_{0}:=\phi^{*}(K_{X}+\Delta)$. We know that $\Delta_{0}$ is effective. If $F$ is a prime divisor on $X_{0}$, since $-(K_{X_{0}}+\Delta_{0})$ is nef and big, then $F$ is dreamy over $(X,\Delta)$ (see [BCHM10, Corollary 1.3.2] for example). Assume that $F$ is exceptional over $X_{0}$. Let $\pi\colon\tilde{X}=X_{m}\to\dots\to X_{1}\to X_{0}$ be the sequence of monoidal transforms with respects to $F$. Since $\pi^{*}\phi^{*}L-A_{(X,\Delta)}(F)F$ is big, by Lemma 3.3 (3),

[TABLE]

is also big. By Lemma 3.3 (2), we have

[TABLE]

This implies that $-K_{\tilde{X}}$ is big. Since $\tilde{X}$ is rational (see [Nak07] for example), the variety $\tilde{X}$ is a Mori dream space in the sense of [HK00] by [TVAV11, Theorem 1]. Thus $F$ is dreamy over $(X,\Delta)$. ∎

Let us recall the following:

Theorem 3.7**.**

Let $(X,\Delta)$ be a log Fano pair and let $F$ be a primitive prime divisor over $X$. Let $\sigma\colon Y\to X$ be the extraction of $F$. Assume that $F$ is not plt-type over $(X,\Delta)$. Then there exists a plt-type prime divisor $G$ over $(X,\Delta)$ such that $c_{X}(G)\subset c_{X}(F)$ and $\hat{\beta}_{(X,\Delta)}(G)<\hat{\beta}_{(X,\Delta)}(F)$.

Proof.

Follows directly from [Fuj17a, Theorem 3.1 and Corollary 3.2]. ∎

Finally, we give an example of non-dreamy prime divisor over $\mathbb{P}^{2}$.

Example 3.8**.**

Assume that $\Bbbk$ is uncountable. Set $X:=\mathbb{P}^{2}$ and $L:=-K_{X}$. Fix any smooth cubic curve $B\subset X$. Take a very general point $p_{1}\in B$ with respects to the inflection points. Then we have $\mathcal{O}_{X}(i)|_{B}\otimes\mathcal{O}_{B}(-3ip_{1})\not\simeq\mathcal{O}_{B}$ for any $i\in\mathbb{Z}_{>0}$. Let us consider the sequence of monoidal transforms $\pi\colon\tilde{X}=X_{9}\to\cdots\to X_{1}\to X_{0}=X$ obtained by:

• •

$X_{0}:=X$, $\pi_{1}\colon X_{1}\to X_{0}$ is the blowup along $p_{1}$ and $E_{1}\subset X_{1}$ is the $\pi_{1}$-exceptional curve.

• •

For $1\leq i\leq 8$, let $p_{i+1}\in X_{i}$ be the intersection of $E_{i}$ and $B^{X_{i}}$. $\pi_{i+1}\colon X_{i+1}\to X_{i}$ is the blowup along $p_{i+1}$ and $E_{i+1}\subset X_{i+1}$ is the $\pi_{i+1}$-exceptional curve.

Let $\tilde{E}_{i}\subset\tilde{X}$ (resp., $\tilde{B}$) be the strict transform of $E_{i}$ (resp., $B$) on $\tilde{X}$ as in Definition 3.1. Since $\tilde{E}_{1},\dots,\tilde{E}_{8}$ are $(-2)$-curves and their dual graph is a straight chain, by [KM98, Proposition 4.10], the morphism $\pi\colon\tilde{X}\to X$ decomposes into

[TABLE]

such that the set of $\nu$-exceptional divisors is equal to the set $\{\tilde{E}_{i}\}_{1\leq i\leq 8}$. Set $F:=\nu_{*}E_{9}$. Obviously, the morphism $\sigma$ is the extraction of $F$ and $\pi$ is the sequence of monoidal transforms with respects to $F$. Moreover, by [KM98, Theorem 4.15], $F$ is plt-type over $X$. Since $E_{9}^{*}=\sum_{i=1}^{9}i\tilde{E}_{i}$, we have

[TABLE]

by Lemma 3.3 (4). Moreover, since $\tilde{B}=\pi^{*}B-\sum_{i=1}^{9}i\tilde{E}_{i}$, we have $\tilde{B}\sim_{\mathbb{Q}}\nu^{*}(\sigma^{*}L-9F)$. Since $\tilde{B}$ is an irreducible curve and $\left(\tilde{B}^{\cdot 2}\right)=0$, the divisor $\sigma^{*}L-9F$ is nef and non-big. Thus we have $\varepsilon(F)=\tau(F)=9$.

Assume that $F$ is dreamy over $X$. Then, as in Definition 2.8, $\sigma^{*}L-9F$ is semiample. Thus,

[TABLE]

is also semiample. This leads to a contradiction. Thus $F$ is non-dreamy over $X$.

4. Product-type prime divisors

4.1. Over $\mathbb{P}^{2}$

In this section, let $C\subset\mathbb{P}^{2}$ be a smooth conic, and let us take $\delta\in[0,1)\cap\mathbb{Q}$ and set $\Delta:=\delta C$. It is well-known that, any $1$-parameter subgroup of $\operatorname{Aut}(\mathbb{P}^{2})=\operatorname{PGL}(3)$ is, after a coordinate change of $\mathbb{P}^{2}$, of the form

[TABLE]

for some $(a^{\prime}$, $b^{\prime})\in\mathbb{Z}^{2}_{\geq 0}\setminus\{(0$, $0)\}$ with $a^{\prime}\geq b^{\prime}$. Let $g\in\mathbb{Z}_{>0}$ be the greatest common factor of $a^{\prime}$ and $b^{\prime}$, and let us set $a:=a^{\prime}/g$, $b:=b^{\prime}/g$. As in [Fuj17b, Example 3.6] (see also [JM12]), the divisorial valuation $v$ on $\Bbbk(\mathbb{P}^{2})$ associates to $\rho$ is the quasi-monomial valuation on

[TABLE]

for coordinates $(x_{1}$, $x_{2})$ (where $x_{i}:=z_{i}/z_{0}$) with weights $(a^{\prime}$, $b^{\prime})$. If $b^{\prime}=0$, then $v=a^{\prime}\cdot\operatorname{ord}_{l}$, where $l\subset\mathbb{P}^{2}$ is the line $(z_{1}=0)$. Assume that $b^{\prime}\geq 1$. Let $\Sigma^{(a,b)}$ (resp., $\Sigma$) be the complete fan in $N_{\mathbb{R}}$ such that the set of $1$-dimensional cones is equal to

[TABLE]

Set $Y^{(a,b)}:=X_{\Sigma^{(a,b)}}$ and let $\sigma\colon Y^{(a,b)}\to X_{\Sigma}=\mathbb{P}^{2}$ be the natural toric morphism. Let $F^{(a,b)}\subset Y^{(a,b)}$ be the $\sigma$-exceptional divisor. Then we know that $v=g\cdot\operatorname{ord}_{F^{(a,b)}}$.

Consequently, we have proved the following:

Lemma 4.1**.**

A prime divisor $F$ over $\mathbb{P}^{2}$ is product-type over $\mathbb{P}^{2}$ if and only if $F$ is a line on $\mathbb{P}^{2}$ or, after a coordinate change of $\mathbb{P}^{2}$, $F$ is equal to the above $F^{(a,b)}$ for some $a$, $b\in\mathbb{Z}_{>0}$ with $a$, $b$ mutually prime.

We consider product-type prime divisor over $(\mathbb{P}^{2},\Delta)$. Take any point $p_{1}\in C$. Let $\pi_{1}\colon X_{1}\to\mathbb{P}^{2}$ be the blowup along $p_{1}$ and let $E_{1}\subset X_{1}$ be the $\pi_{1}$-exceptional curve. Let $p_{2}\in X_{1}$ be the intersection of $E_{1}$ and $C^{X_{1}}$. Let $\pi_{2}\colon X_{2}\to X_{1}$ be the blowup along $p_{2}$ and let $E_{2}\subset X_{2}$ be the $\pi_{2}$-exceptional curve.

Proposition 4.2**.**

The above $E_{2}$ is a product-type prime divisor over $(\mathbb{P}^{2},\Delta)$.

Proof.

We may assume that $p_{1}=(1:0:0)$ and $C=(z_{2}^{2}+z_{0}z_{1}=0)\subset\mathbb{P}^{2}_{z_{0}:z_{1}:z_{2}}$. Then, as we have seen in Lemma 4.1, the divisorial valuation $\operatorname{ord}_{E_{2}}\colon\Bbbk(\mathbb{P}^{2})^{*}\to\mathbb{Z}$ corresponds to the $1$-parameter subgroup

[TABLE]

For any $t\in\mathbb{G}_{m}$, we have $\rho_{t}^{*}C=C$. Thus $\rho$ factors through

[TABLE]

Hence $E_{2}$ is a product-type prime divisor over $(\mathbb{P}^{2},\Delta)$. ∎

4.2. Over $\mathbb{P}^{1}\times\mathbb{P}^{1}$

In this section, let $C\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ be the diagonal, and let $\delta\in[0$, $1)\cap\mathbb{Q}$, and set $\Delta:=\delta C$. Let us consider the 1-parameter subgroup

[TABLE]

Since $C\subset\mathbb{P}^{1}_{z_{10}:z_{11}}\times\mathbb{P}^{1}_{z_{20}:z_{21}}$ is defined by the equation $z_{10}z_{21}=z_{11}z_{20}$, we have $\rho_{t}^{*}C=C$ for any $t\in\mathbb{G}_{m}$. Thus $\rho$ factors through $\operatorname{Aut}(\mathbb{P}^{1}\times\mathbb{P}^{1},\Delta)$. On the other hand, the morphism

[TABLE]

induces the inclusion

[TABLE]

where $x_{i}:=z_{i1}/z_{i0}$ for $i=1$, $2$. Thus, as in [Fuj17b, §3], the divisorial valuation $v$ on $\Bbbk(\mathbb{P}^{1}\times\mathbb{P}^{1})$ associates to $\rho$ is the quasi-monomial valuation on $\mathbb{A}^{2}_{x_{1},x_{2}}=\mathbb{P}^{1}\times\mathbb{P}^{1}\setminus(z_{10}z_{20}=0)$ for coordinates $(x_{1}$, $x_{2})$ with weights $(1$, $1)$. In other words, if $F$ is the exceptional divisor of the ordinary blowup of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ along $(1:0;1:0)$, then $v$ is equal to $\operatorname{ord}_{F}$. Thus we have proved the following proposition:

Proposition 4.3**.**

Let $F$ be the exceptional divisor of the ordinary blowup of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ along a point on $C$. Then $F$ is product-type over $(\mathbb{P}^{1}\times\mathbb{P}^{1},\Delta)$.

In order to prove Theorem 1.1 (2), we need the following lemma:

Lemma 4.4**.**

The divisor $C$ on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ is not a product-type prime divisor over $(\mathbb{P}^{1}\times\mathbb{P}^{1},\Delta)$.

Proof.

Assume not. Then, as in the proof in [Fuj17b, Lemma 3.8], $\mathbb{P}^{1}\times\mathbb{P}^{1}$ must be isomorphic to

[TABLE]

where $S_{k,j}:=\mathcal{F}^{j}V_{k}/\mathcal{F}^{j+1}V_{k}$ with

[TABLE]

Note that

[TABLE]

Thus we have the natural isomorphism

[TABLE]

In particular, the variety $\mathcal{X}_{0}$ is isomorphic to the weighted projective plane $\mathbb{P}(1,1,2)$ of weights $(1,1,2)$. This leads to a contradiction. ∎

5. On the projective plane

In this section, we set $X:=\mathbb{P}^{2}$, let $C\subset X$ be a smooth conic, fix $\delta\in[0,1)\cap\mathbb{Q}$ and set $\Delta:=\delta C$. We prove the following theorem. Theorem 1.1 (1) is an immediate consequence of Theorem 5.1.

Theorem 5.1**.**

1. (1)

*If $\delta>3/4$ *$($resp., $\delta\geq 3/4)$, then $(X,\Delta)$ is not K-semistable $($resp., not K-polystable$)$. 2. (2)

The pair $(X,\Delta)$ is no longer K-stable for any $\delta\in[0,1)\cap\mathbb{Q}$. 3. (3)

Assume that $\delta\leq 3/4$. For any prime divisor $F$ over $X$, we have $\hat{\beta}_{(X,\Delta)}(F)\geq 0$. 4. (4)

If $\delta<3/4$ and if a prime divisor $F$ over $X$ satisfies that $\hat{\beta}_{(X,\Delta)}(F)=0$, then $F$ is a product-type prime divisor over $(X,\Delta)$.

Proof.

The proof is based on the ideas in [Fuj17a, §4.2]. However, we need more delicate arguments.

Step 1. Take any prime divisor $F$ on $X$. Set $d:=\deg F$. If $F\neq C$, then we have

[TABLE]

Moreover, equality holds if and only if $d=1$ and $\delta=0$. We already know in Lemma 4.1 that a line is product-type prime divisor over $X$. If $F=C$, then $d=2$ and $A_{(X,\Delta)}(C)=1-\delta$. Thus we have

[TABLE]

By Lemma 4.1, $C$ is not a product-type prime divisor over $(X,\Delta)$. Thus we have proved Theorem 5.1 (1).

Step 2. Let us prove Theorem 5.1 (2), (3) and (4). From now on, we assume that $\delta\leq 3/4$. Let $F$ be a prime divisor over $X$. By Step 1, Proposition 3.6 and Theorem 3.7, we may assume that $F$ is exceptional over $X$, dreamy over $(X,\Delta)$ and plt-type over $(X,\Delta)$. Of course, $F$ is plt-type over $X$. Let $\pi\colon\tilde{X}=X_{m}\to\dots\to X_{1}\to X_{0}=X$, $E_{i}^{*}$ $(1\leq i\leq m)$, $a^{F}$, $b^{F}$, $f$, etc., be as in Definitions 3.1, 3.2 and 3.4. Moreover, let us set $a:=a^{F}$, $b:=b^{F}$, $\varepsilon:=\varepsilon(F)$, $\tau:=\tau(F)$, $A_{i}:=A_{(X,\Delta)}(E_{i})$ $(1\leq i\leq m)$ and $A:=A_{(X,\Delta)}(F)$ for simplicity.

Let $l_{1}\subset X$ be a general line passing through $p_{1}$. By Lemmas 2.9, 3.3 and 3.5, we have

[TABLE]

Thus we get $\tau\leq a(3-2\delta)$ and $\varepsilon+\tau\leq(a+b)(3-2\delta)$ (recall that $\varepsilon\tau=ab(3-2\delta)^{2}$ by Lemma 2.10).

Step 3. We consider the case $m=1$, i.e., $a=b=1$. Then, since $Y=\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}\oplus\mathcal{O}(1))$, we have $\varepsilon=\tau=3-2\delta$. If $p_{1}\not\in C$, then we have $A=2$; if $p_{1}\in C$, then we have $A=2-\delta$. By Lemma 2.10, we get

[TABLE]

Hence we have $\hat{\beta}_{(X,\Delta)}(F)\geq 0$. If $\hat{\beta}_{(X,\Delta)}(F)=0$, then $\delta=0$ and $F$ is a product-type prime divisor over $(X,\Delta)\,(=X)$ by Lemma 4.1.

Step 4. Thus we may further assume that $m\geq 2$. Let $l_{0}\subset X$ be the unique line such that $l_{0}^{X_{2}}\cap E_{2}\neq\emptyset$. Let us set

[TABLE]

where $k=\lceil(a/b)\rceil$ as in Definition 3.4.

We consider the case $p_{2}\not\in C^{X_{1}}$. If $p_{1}\in C$ and $\delta>0$, then we have $A_{1}=2-\delta$ and

[TABLE]

Thus we can inductively show that $A_{i}=a_{i}+b_{i}-b_{i}\delta$. In particular, we have $A=a+b-b\delta$. By Lemma 2.10 and Step 2, we have

[TABLE]

If $p_{1}\not\in C$, then we have $A=a+b$. By Lemma 2.10 and Step 2, we have

[TABLE]

by Step 2. Assume that $\hat{\beta}_{(X,\Delta)}(F)=0$. Then $\delta=0$ and $(\varepsilon$, $\tau)=(3b$, $3a)$. Thus $\mu\colon Y\to Z$ is birational, where $\mu$ is as in Definition 2.8. If $l_{0}^{Y}$ is not $\mu$-exceptional, then, by Lemma 2.9, we have

[TABLE]

However, by Lemma 3.5, we have $\operatorname{coeff}_{\tilde{E}_{j_{0}}}E_{m}^{*}>b$. This leads to a contradiction. Thus $l_{0}^{Y}$ is $\mu$-exceptional. By Lemma 2.9, we have

[TABLE]

Thus we have $\operatorname{coeff}_{\tilde{E}_{j_{0}}}E_{m}^{*}=a$. This implies that $j_{0}=k$. As we have already seen in Lemma 4.1, $F$ is a product-type prime divisor over $(X,\Delta)\,(=X)$.

Step 5. Thus we may further assume that $\delta>0$ and $p_{2}\in C^{X_{1}}$. In this case, $l_{0}^{X_{1}}$ and $C^{X_{1}}$ intersect transversally at $p_{2}$.

We consider the case $m=2$. In this case, $F$ is product-type over $(X,\Delta)$ by Proposition 4.2. Since $\tilde{X}$ is toric, we can easily show that $l_{0}^{Y}$ is the unique $\mu$-exceptional curve. By Lemma 2.9, we have

[TABLE]

Thus we have $(\varepsilon$, $\tau)=(3-2\delta$, $2(3-2\delta))$, and $\hat{\beta}_{(X,\Delta)}(F)=0$ by Lemma 2.10 (note that $A=3-2\delta$). In particular, we have proved Theorem 5.1 (2).

Step 6. Thus we may further assume that $m\geq 3$. We consider the case $p_{3}\in E_{1}^{X_{2}}$. In this case, we have $k=2$. Thus $2b>a$ holds. Since $A_{1}=2-\delta$, $A_{2}=3-2\delta$ and $A_{i}=A_{q(i)}+A_{i-1}$ for any $3\leq i\leq m$, we can inductively show that $A_{i}=a_{i}+b_{i}-a_{i}\delta$. In particular, we have $A=a+b-a\delta$. By Lemma 2.10 and Step 2, we get

[TABLE]

Step 7. Thus we may further assume that $p_{3}\not\in E_{1}^{X_{2}}$. Then $k\geq 3$. In particular, we have $a>2b$. Let us set

[TABLE]

Since $l_{0}^{X_{2}}\cap C^{X_{2}}=\emptyset$, either $j_{0}$ or $j_{C}$ is equal to $2$. By the definitions of $j_{C}$ and $k$, we have

[TABLE]

Therefore, we can inductively show that, for any $k\leq i\leq m$,

[TABLE]

In particular, we have $A=a+b-\min\{j_{C}b$, $a\}\delta$.

Assume that $j_{C}=2$. Then $A=a+b-2b\delta$. By Lemma 2.10 and Step 2, we get

[TABLE]

Step 8. Thus we may further assume that $j_{C}\geq 3$. This implies that $j_{0}=2$. If $l_{0}^{Y}$ is $\mu$-exceptional, then we have

[TABLE]

by Lemma 2.9. Thus we get $\varepsilon+\tau=(a/2+2b)(3-2\delta)$. If $l_{0}^{Y}$ is not $\mu$-exceptional, then we have

[TABLE]

by Lemma 2.9. Thus, in any case, we have the inequality

[TABLE]

By Lemma 2.10, we have

[TABLE]

Moreover, if $\delta<3/4$, then $\hat{\beta}_{(X,\Delta)}(F)>0$.

As a consequence, we have completed the proof of Theorem 5.1. ∎

Remark 5.2**.**

One may expects that there might be a positive constant $\varepsilon_{0}$ such that $\hat{\beta}_{X}(F)\geq\varepsilon_{0}$ holds for any non-product-type prime divisor $F$ over $X=\mathbb{P}^{2}$. However, this is not true. See the following example.

Example 5.3**.**

Let $l\subset X$ be a line. Fix any $m\geq 4$. Take any point $p_{1}\in l$ and let us consider the sequence of monoidal transforms $\pi^{\prime}\colon X_{m-1}\to\cdots\to X_{1}\to X_{0}$ obtained by:

• •

$X_{0}:=X$, $\pi_{1}\colon X_{1}\to X_{0}$ is the blowup along $p_{1}$ and let $E_{1}\subset X_{1}$ be the $\pi_{1}$-exceptional curve.

• •

For any $2\leq i\leq m-2$, let $p_{i+1}\in X_{i}$ be the intersection of $l^{X_{i}}$ and $E_{i}$, let $\pi_{i+1}\colon X_{i+1}\to X_{i}$ be the blowup along $p_{i+1}$, and let $E_{i+1}\subset X_{i+1}$ be the $\pi_{i+1}$-exceptional curve.

Moreover, let us take $p_{m}\in E_{m-1}$ with $p_{m}\not\in l^{X_{m-1}}\cup E_{m-2}^{X_{m-1}}$, let $\pi_{m}\colon X_{m}\to X_{m-1}$ be the blowup along $p_{m}$ and let $E_{m}\subset X_{m}$ be the $\pi_{m}$-exceptional curve. Set $\pi:=\pi^{\prime}\circ\pi_{m}$, $\tilde{X}:=X_{m}$ and let $\tilde{E}_{i}$ (resp., $\tilde{l}$) be the strict transform of $E_{i}$ (resp., $l$) on $\tilde{X}$. Then the dual graph of $\tilde{E}_{1},\dots,\tilde{E}_{m},\tilde{l}$ is the following:

\tilde{E}_{1}$$\tilde{E}_{m-1}$$\tilde{l}$$\tilde{E}_{m}

Note that $\left(\tilde{E}_{i}^{\cdot 2}\right)=-2$ for $1\leq i\leq m-1$ and $\left(\tilde{l}^{\cdot 2}\right)=-(m-2)$. By [KM98, Proposition 4.10], the morphism $\pi$ decomposes into

[TABLE]

such that the set of $\nu$-exceptional divisors on $\tilde{X}$ is equal to the set $\{\tilde{E}_{i}\}_{1\leq i\leq m-1}$. Obviously, $F:=\nu_{*}\tilde{E}_{m}$ is primitive over $X$ and $F$ is plt-type over $X$ by [KM98, Theorem 4.15]. Again by [KM98, Proposition 4.10], we can contract $\tilde{E}_{1},\dots,\tilde{E}_{m-1}$, $\tilde{l}$. In particular, there exists a birational morphism $\mu\colon Y\to Z$ such that $l^{Y}$ is the unique $\mu$-exceptional curve. This implies that $F$ is dreamy over $X$ (see [HK00] for example) and the standard diagram with respects to $F$ consists of $\sigma$ and $\mu$.

From the construction, we have $a^{F}=m$, $b^{F}=1$, $f=m$, $A_{X}(F)=m+1$. Moreover, by Lemma 2.9, we have the equality

[TABLE]

By Lemma 3.3, we have $\varepsilon(F)=3m/(m-1)$. By Lemma 2.10, we get

[TABLE]

Therefore, we have $\hat{\beta}_{X}(F)\searrow 0$ when $m\to\infty$.

6. On the product of the projective lines

In this section, we set $X:=\mathbb{P}^{1}\times\mathbb{P}^{1}$, let $C\subset X$ be the diagonal, fix $\delta\in(0,1)\cap\mathbb{Q}$ and set $\Delta:=\delta C$. We recall the following result:

Theorem 6.1** (see [Kem78, Li17, Blu16, BJ17]).**

$X=\mathbb{P}^{1}\times\mathbb{P}^{1}$* is K-semistable.*

Remark 6.2**.**

When $\Bbbk=\mathbb{C}$, the above result is well-known (see [Tia97, Don02]). We emphasize that some proofs of Theorem 6.1 are purely algebraic. When $\Bbbk=\mathbb{C}$, the K-polystability of $X$ is also known (see [Ber16]). Moreover, recently, K-polystability of $X$ was proved purely algebraically by [LWX18].

In this section, we algebraically prove the following theorem by using Theorem 6.1. Theorem 1.1 (2) is an immediate consequence of Theorem 6.3.

Theorem 6.3**.**

1. (1)

*If $\delta>1/2$ *$($resp., $\delta\geq 1/2)$, then $(X,\Delta)$ is not K-semistable $($resp., not K-polystable$)$. 2. (2)

The pair $(X,\Delta)$ is no longer K-stable for any $\delta\in(0,1)\cap\mathbb{Q}$. 3. (3)

Assume that $\delta\leq 1/2$. For any prime divisor $F$ over $X$, we have $\hat{\beta}_{(X,\Delta)}(F)\geq 0$. 4. (4)

If $\delta<1/2$ and if a prime divisor $F$ over $X$ satisfies that $\hat{\beta}_{(X,\Delta)}(F)=0$, then $F$ is a product-type prime divisor over $(X,\Delta)$.

Proof.

The proof is similar to the proof of Theorem 5.1. The proof of Theorem 6.3 is more complicated than the proof of Theorem 5.1.

Step 1. Since $A_{(X,\Delta)}(C)=1-\delta$, we have

[TABLE]

By Lemma 4.4, we have proved Theorem 6.3 (1). We may assume that $\delta\leq 1/2$.

Step 2. Take any prime divisor $F$ over $X$. By Theorem 6.1, we have

[TABLE]

Thus, if $A_{X}(F)=A_{(X,\Delta)}(F)$, i.e., if $c_{X}(F)\not\subset C$ holds, then we have the inequality $\hat{\beta}_{(X,\Delta)}(F)\geq\delta/2>0$.

Step 3. Thus we may assume that $F$ is exceptional over $X$ and $c_{X}(F)\in C$. Moreover, by Proposition 3.6 and Theorem 3.7, we may assume that $F$ is dreamy over $(X,\Delta)$ and plt-type over $(X,\Delta)$. Let $l_{1}$, $l_{2}\subset X$ be the fibers of the fibrations $\mathbb{P}^{1}_{z_{10}:z_{11}}\times\mathbb{P}^{1}_{z_{20}:z_{21}}\to\mathbb{P}^{1}_{z_{10}:z_{11}}$ and $\mathbb{P}^{1}_{z_{10}:z_{11}}\times\mathbb{P}^{1}_{z_{20}:z_{21}}\to\mathbb{P}^{1}_{z_{20}:z_{21}}$ passing through $c_{X}(F)$. Let $\pi\colon\tilde{X}=X_{m}\to\dots\to X_{1}\to X_{0}=X$, $E_{i}^{*}$ $(1\leq i\leq m)$, $a^{F}$, $b^{F}$, $f$, etc., be as in Definitions 3.1, 3.2 and 3.4. Moreover, let us set $a:=a^{F}$, $b:=b^{F}$, $\varepsilon:=\varepsilon(F)$, $A_{i}:=A_{(X,\Delta)}(E_{i})$ $(1\leq i\leq m)$ and $A:=A_{(X,\Delta)}(F)$ for simplicity. For any $x\in[0$, $\varepsilon]$, we have

[TABLE]

From Proposition 2.7, we have

[TABLE]

Therefore we get the inequality

[TABLE]

Step 4. We consider the case $m=1$, i.e., $a=b=1$ and $A=2-\delta$. Since $Y$ is the del Pezzo surface of degree $7$, we can easily show that

[TABLE]

Thus we get the equality $\hat{\beta}_{(X,\Delta)}(F)=0$. In fact, by Proposition 4.3, the divisor $F$ is a product-type prime divisor over $(X,\Delta)$. In particular, we have proved Theorem 6.3 (2).

Step 5. We consider the case $m\geq 2$. Assume that $p_{2}\not\in C^{X_{1}}$. Then we can inductively show that $A_{i}=a_{i}+b_{i}-b_{i}\delta$. In particular, we have $A=a+b-b\delta$. By Step 2, we have

[TABLE]

Thus we may assume that $m\geq 2$ and $p_{2}\in C^{X_{1}}$.

Step 6. Let us set

[TABLE]

Then we can inductively show that

[TABLE]

As in the argument in Step 7 for the proof of Theorem 5.1, we have $A=a+b-\min\{j_{C}b$, $a\}\delta$. If $2j_{C}\leq k$, then, from Step 2, we have

[TABLE]

Thus we may further assume that $2j_{C}>k$.

Step 7. Assume that $k=2$. Then we have $2b\geq a$ and $A=a+b-a\delta$. By Lemma 3.5 (4), we have

[TABLE]

Thus we have $\varepsilon\leq a(2-\delta)$. Since $C^{\tilde{X}}$ is nef and $\pi^{*}C=C^{\tilde{X}}+\sum_{i=1}^{m}a_{i}\tilde{E}_{i}$, we have $\varepsilon=a(2-\delta)$ by Lemma 3.3 (5). By Step 3, we get

[TABLE]

When $\delta<1/2$, then we get $\hat{\beta}_{(X,\Delta)}(F)>0$.

Step 8. Thus we may further assume that $k\geq 3$. By Lemma 3.5 (4), we have

[TABLE]

Thus we get

[TABLE]

Assume that $j_{C}^{2}b\geq 2a$. Then, by the assumption $k\geq 3$, Lemma 3.3 (5) and Step 6, we have

[TABLE]

Therefore, by Step 3, we have

[TABLE]

If $j_{C}=k$, then we have

[TABLE]

When $j_{C}=k$ and $\delta<1/2$, we have $\hat{\beta}_{(X,\Delta)}(F)>0$. If $j_{C}<k$, then we have

[TABLE]

Step 9. Thus we can further assume that $j_{C}^{2}b<2a$. Since we have already assumed that $2j_{C}>k$, we get

[TABLE]

This implies that $(j_{C}$, $k)=(2$, $3)$ or $(3$, $5)$. Moreover, if $(j_{C}$, $k)=(3$, $5)$, then we may assume that $a/b>9/2$. Let $\lambda\colon X_{1}\to\mathbb{P}^{2}$ be the birational morphism contracting $l_{1}^{X_{1}}$ and $l_{2}^{X_{1}}$.

Step 10. Assume that $(j_{C}$, $k)=(2$, $3)$. Then we can uniquely find the line $l$ on $\mathbb{P}^{2}$ with $l^{X_{3}}\cap E_{3}\neq\emptyset$. Since $a\leq 3b$, we have

[TABLE]

by Lemma 3.5 (4). Thus we have $\varepsilon\leq 2b(2-\delta)$. Moreover, we can inductively show that

[TABLE]

By Lemma 3.3 (5), we get $\varepsilon=2b(2-\delta)$. Thus, from Step 3, we have

[TABLE]

Step 11. Assume that $(j_{C}$, $k)=(3$, $5)$ and $a/b>9/2$. It is well-known that there exists a unique smooth conic $D$ on $\mathbb{P}^{2}$ such that $D^{X_{5}}\cap l_{1}^{X_{5}}\neq\emptyset$ and $D^{X_{5}}\cap E_{5}\neq\emptyset$. By Lemma 3.5 (4), we have

[TABLE]

Thus we get $\varepsilon\leq 3b(2-\delta)$. On the other hand, we know that

[TABLE]

and

[TABLE]

Thus we get $\varepsilon=3b(2-\delta)$ by Lemma 3.3 (5). Hence we have

[TABLE]

As a consequence, we have completed the proof of Theorem 6.3. ∎

7. Proof of Corollary 1.2

In this section, we prove Corollary 1.2. We recall the result of Maeda. We set $\mathbb{F}_{m}:=\mathbb{P}_{\mathbb{P}^{1}}\left(\mathcal{O}\oplus\mathcal{O}(m)\right)$ ($m\geq 0$) and let $e\subset\mathbb{F}_{m}$ be a section of $\mathbb{F}_{m}\to\mathbb{P}^{1}$ with the self intersection number $-m$, let $l\subset\mathbb{F}_{m}$ be a fiber of $\mathbb{F}_{m}\to\mathbb{P}^{1}$, and let $e_{\infty}\subset\mathbb{F}_{m}$ is a section of $\mathbb{F}_{m}\to\mathbb{P}^{1}$ with the self intersection number $m$.

Theorem 7.1** ([Mae86]).**

Let $X$ be a smooth projective surface and let $D$ be a nonzero effective reduced simple normal crossing divisor on $X$ with $-(K_{X}+D)$ ample. Then $(X,D)$ is isomorphic to one of $(\mathbb{P}^{2}$, line$)$, $(\mathbb{P}^{2}$, the union of two distinct lines$)$, $(\mathbb{P}^{2}$, smooth conic$)$, $(\mathbb{P}^{1}\times\mathbb{P}^{1}$, diagonal$)$, $(\mathbb{F}_{1}$, $e_{\infty})$, $(\mathbb{F}_{m}$, $e)$, or $(\mathbb{F}_{m}$, $e+l)$.

Corollary 1.2 is an immediate consequence of Theorems 5.1, 6.3, 7.1 and [BB13, Theorem 1.2] for example. We give an elemental proof of Corollary 1.2 for the readers’ convenience.

Proof of Corollary 1.2.

Assume that $X=\mathbb{F}_{m}$ and $\Delta=\delta_{1}e+\delta_{2}l$ with $\delta_{1}\in(0$, $1)\cap\mathbb{Q}$ and $\delta_{2}\in[0$, $1)\cap\mathbb{Q}$. Then the pair $(X,\Delta)$ is a log del Pezzo pair if and only if $m+2-\delta_{2}>m(2-\delta_{1})$. The $\mathbb{R}$-divisor $L-xe\sim_{\mathbb{R}}(2-\delta_{1}-x)e+(m+2-\delta_{2})l$ for $x\in\mathbb{R}_{\geq 0}$ is nef if and only if $x\leq 2-\delta_{1}$. Thus we have

[TABLE]

If $m=0$, then we can immediately show that $\hat{\beta}_{(X,\Delta)}(e)<0$; if $m\geq 1$, then $\hat{\beta}_{(X,\Delta)}(e)<0$ since

[TABLE]

Assume that $X=\mathbb{F}_{1}$ and $\Delta=\delta e_{\infty}$ with $\delta\in[0$, $1)\cap\mathbb{Q}$. The $\mathbb{R}$-divisor $L-xe\sim_{\mathbb{R}}(2-\delta-x)e+(3-\delta)l$ for $x\in\mathbb{R}_{\geq 0}$ is nef if and only if $x\leq 2-\delta$. Thus we have

[TABLE]

If $\delta<2-\sqrt{3}$, then $\hat{\beta}_{(X,\Delta)}(e)<0$. Similarly, The $\mathbb{R}$-divisor $L-xe_{\infty}\sim_{\mathbb{R}}(2-\delta-x)e+(3-\delta-x)l$ for $x\in\mathbb{R}_{\geq 0}$ is nef if and only if $x\leq 2-\delta$. Thus we have

[TABLE]

If $\delta>2-\sqrt{3}$, then we have $\hat{\beta}_{(X,\Delta)}(e_{\infty})<0$. Since $\delta\in\mathbb{Q}$, the pair $(X,\Delta)$ is not K-semistable for any $\delta\in[0$, $1)\cap\mathbb{Q}$.

Assume that $X=\mathbb{P}^{2}$ and $\Delta=\delta_{1}l_{1}+\delta_{2}l_{2}$ with $l_{1}$, $l_{2}$ distinct lines, $\delta_{1}$, $\delta_{2}\in[0$, $1)\cap\mathbb{Q}$, $\delta_{1}\leq\delta_{2}$ and $(\delta_{1}$, $\delta_{2})\neq(0$, $0)$. Then we can immediately get the inequality

[TABLE]

Together with Theorems 5.1 and 6.3, we get the assertion. ∎