K-STABLE DIVISORS IN $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^2$ OF DEGREE $(1,1,2)$

Abstract

We prove that every smooth divisor in $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^2$ of degree $(1,1,2)$ is K-stable.

1 Introduction

Smooth Fano threefolds have been classified by Iskovskikh, Mori, and Mukai into $105$ families, which are labeled as

1.1,

1.2,

1.3, $\ldots $ ,

10.1. See [3] for the description of these families. Threefolds in each of these $105$ deformation families can be parametrized by a nonempty rational irreducible variety. It has been proved in [3], [11], [12] that the deformation families

do not have smooth K-polystable members, and general members of the remaining 78 deformation families are K-polystable. In fact, for 54 among these 78 families, we know all K-polystable smooth members [2]–[6], [9], [14], [16]. The remaining $24$ deformation families are

The goal of this paper is to show that all smooth Fano threefolds in the family 3.3 are K-stable. Smooth members of this deformation family are smooth divisors in $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^2$ of degree $(1,1,2)$ . To be precise, we prove the following result.

Main Theorem. Let X be a smooth divisor in $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^2$ of degree $(1,1,2)$ . Then X is K-stable.

2 Smooth Fano threefolds in the deformation family 3.3

Let X be a divisor in $\mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}\times \mathbb {P}^2_{x,y,z}$ of tridegree $(1,1,2)$ , where $([s:t],[u:v],[x:y:z])$ are coordinates on $\mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}\times \mathbb {P}^2_{x,y,z}$ . Then X is given by the following equation:

$$ \begin{align*}\left[ \begin{array}{cc} s & t\\ \end{array} \right] \left[ \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{array} \right] \left[ \begin{array}{c} u \\ v \\ \end{array} \right]=0, \end{align*} $$

where each $a_{ij}=a_{ij}(x,y,z)$ is a homogeneous polynomials of degree $2$ . We can also define X by

$$ \begin{align*}\left[ \begin{array}{ccc} x & y & z\\ \end{array} \right] \left[ \begin{array}{ccc} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23}\\ b_{31} & b_{32} & b_{33}\\ \end{array} \right] \left[ \begin{array}{c} x \\ y \\ z \\ \end{array} \right]=0, \end{align*} $$

where each $b_{ij}=b_{ij}(s,t;u,v)$ is a bi-homogeneous polynomial of degree $(1,1)$ .

Suppose that X is smooth. Then X is a smooth Fano threefold in the deformation family

3.3. Moreover, every smooth Fano threefold in this deformation family can be obtained in this way. Observe that $-K_X^3=18$ , and we have the following commutative diagram:

where all maps are induced by natural projections. Note that $\omega $ is a (standard) conic bundle whose discriminant curve $\Delta _{\mathbb {P}^1\times \mathbb {P}^1}\subset \mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}$ is a (possibly singular) curve of degree $(3,3)$ given by

$$ \begin{align*}\mathrm{det}\left[\begin{array}{ccc} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23}\\ b_{31} & b_{32} & b_{33}\\ \end{array} \right]=0. \end{align*} $$

Similarly, the map $\pi _3$ is a (nonstandard) conic bundle whose discriminant curve $\Delta _{\mathbb {P}^2}$ is a smooth plane quartic curve in $\mathbb {P}^2_{x,y,z}$ , which is given by $a_{11}a_{22}=a_{12}a_{21}$ . Both maps $\phi _1$ and $\phi _2$ are birational morphisms that blow up the following smooth genus $3$ curves:

$$ \begin{align*} \big\{sa_{11}+ta_{21}=sa_{12}+ta_{22}=0\big\}&\subset\mathbb{P}^1_{s,t}\times\mathbb{P}^2_{x,y,z},\\ \big\{ua_{11}+va_{12}=ua_{21}+va_{22}=0\big\}&\subset\mathbb{P}^1_{u,v}\times\mathbb{P}^2_{x,y,z}. \end{align*} $$

Finally, both morphisms $\pi _1$ and $\pi _2$ are fibrations into quintic del Pezzo surfaces.

Let $H_1=\pi _1^*(\mathcal {O}_{\mathbb {P}^1}(1))$ , let $H_2=\pi _2^*(\mathcal {O}_{\mathbb {P}^1}(1))$ , let $H_3=\pi _3^*(\mathcal {O}_{\mathbb {P}^2}(1))$ , and let $E_1$ and $E_2$ be the exceptional divisors of the morphisms $\phi _1$ and $\phi _2$ , respectively. Then

$$ \begin{align*} -K_X&\sim H_1+H_2+H_3,\\ E_1&\sim H_1+2H_3-H_2,\\ E_2&\sim H_2+2H_3-H_1. \end{align*} $$

This gives $E_1+E_2\sim 4H_3$ , which also follows from $E_1+E_2=\pi _3^*(\Delta _{\mathbb {P}^2})$ . We have

$$ \begin{align*}-K_X\sim_{\mathbb{Q}} \frac{3}{2}H_1+\frac{1}{2}H_2+\frac{1}{2}E_2\sim_{\mathbb{Q}} \frac{1}{2}H_1+\frac{3}{2}H_2+\frac{1}{2}E_1. \end{align*} $$

In particular, we see that $\alpha (X)\leqslant \frac {2}{3}$ . Note that $E_1\cong E_2\cong \Delta _{\mathbb {P}^2}\times \mathbb {P}^1$ .

The Mori cone $\overline {\mathrm {NE}}(X)$ is simplicial and is generated by the curves contracted by $\omega $ , $\phi _1$ , and $\phi _2$ . The cone of effective divisors $\mathrm {Eff}(X)$ is generated by the classes of the divisors $E_1$ , $E_2$ , $H_1$ , and $H_2$ .

Lemma 1. Let S be a surface in the pencil $|H_1|$ . Then S is a normal quintic del Pezzo surface that has at most Du Val singularities, the restriction $\pi _3\vert _{S}\colon S\to \mathbb {P}^2_{x,y,z}$ is a birational morphism, and the restriction $\pi _2\vert _{S}\colon S\to \mathbb {P}^1_{u,v}$ is a conic bundle. Moreover, one of the following cases holds:

• $\bullet $ The surface S is smooth.

• (𝔸₁) The surface S has one singular point of type $\mathbb {A}_1$ .

• (2𝔸₁) The surface S has two singular points of type $\mathbb {A}_1$ .

• (𝔸₂) The surface S has one singular point of type $\mathbb {A}_2$ .

• (𝔸₃) The surface S has one singular point of type $\mathbb {A}_3$ .

Furthermore, in each of these five cases, the del Pezzo surface S is unique up to an isomorphism.

Proof. This is well known [7], [8].

Remark 2. In the notations and assumptions of Lemma 1, suppose that the surface S is singular, and let $\varpi \colon \widetilde {S}\to S$ be its minimal resolution of singularities. Then the dual graph of the $(-1)$ -curves and $(-2)$ -curves on the surface $\widetilde {S}$ can be described as follows:

• ( $\mathbb {A}_1$ ) if S has one singular point of type $\mathbb {A}_1$ , then the dual graph is

  

• ( $2\mathbb {A}_1$ ) if S has two singular points of type $\mathbb {A}_1$ , then the dual graph is

  

• ( $\mathbb {A}_2$ ) if S has one singular point of type $\mathbb {A}_2$ , then the dual graph is

  

• ( $\mathbb {A}_3$ ) if S has one singular point of type $\mathbb {A}_3$ , then the dual graph is

  

Here, as in the papers [7], [8], we denote a $(-1)$ -curve by $\bullet $ , and we denote a $(-2)$ -curve by $\circ $ .

Lemma 3. Let $S_1$ be a surface in $|H_1|$ , let $S_2$ be a surface in $|H_2|$ , and let P be a point in $S_1\cap S_2$ . Then at least one of the surfaces $S_1$ or $S_2$ is smooth at P.

Proof. Local computations.

Corollary 4. In the notations and assumptions of Lemma 3, suppose that the conic $S_1\cdot S_2$ is reduced. Then at least one of the surfaces $S_1$ or $S_2$ is smooth along $S_1\cap S_2$ .

Lemma 5. Let P be a point in X, let C be the scheme fiber of the conic bundle $\omega $ that contains P, and let Z be the scheme fiber of the conic bundle $\pi _3$ that contains P. Then C or Z is smooth at P.

Proof. Local computations.

Lemma 6. Let C be a fiber of the morphism $\pi _3$ , and let S be a general surface in $|H_3|$ that contains C. Then S is smooth, $K_S^2=4$ , and $-K_S\sim (H_1+H_2)\vert _{S}$ , which implies that $-K_S$ is nef and big. Moreover, one of the following three cases holds:

1. (1) The conic C is smooth, $-K_S$ is ample, and the restriction $\omega \vert _{S}\colon S\to \mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}$ is a double cover branched over a smooth curve of degree $(2,2)$ .

2. (2) The conic C is smooth, the divisor $-K_S$ is not ample, the conic $\omega (C)$ is an irreducible component of the discriminant curve $\Delta _{\mathbb {P}^1\times \mathbb {P}^1}$ , the conic C is contained in $\mathrm {Sing}(\omega ^{-1}(\Delta _{\mathbb {P}^1\times \mathbb {P}^1}))$ , and the restriction map $\omega \vert _{S}\colon S\to \mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}$ fits the following commutative diagram:

   
   where $\alpha $ is a birational morphism that contracts two disjoint $(-2)$ -curves, and $\beta $ is a double cover branched over a singular curve of degree $(2,2)$ , which is a union of the curve $\omega (C)$ and another smooth curve of degree $(1,1)$ , which intersect transversally at two distinct points.

3. (3) The conic C is singular, $-K_S$ is ample, and the restriction $\omega \vert _{S}\colon S\to \mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}$ is a double cover branched over a smooth curve of degree $(2,2)$ .

Proof. The smoothness of the surface S easily follows from local computations. If $-K_S$ is ample, the remaining assertions are obvious. So, to complete the proof, we assume that $-K_S$ is not ample. Then the restriction $\omega \vert _{S}\colon S\to \mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}$ fits the commutative diagram

where $\alpha $ is a birational morphism that contracts all $(-2)$ -curves in S, and $\beta $ is a double cover branched over a singular curve of degree $(2,2)$ . Let $\ell $ be a $(-2)$ -curve in S. Then

$$ \begin{align*}(H_1+H_2)\cdot\ell=-K_S\cdot\ell=0, \end{align*} $$

so that $\omega (\ell )$ is a point in $\mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}$ . But $\pi _3(\ell )$ is a line in $\mathbb {P}^2_{x,y,z}$ that contains the point $\pi _3(C)$ . This shows that the curve $\ell $ is an irreducible component of a singular fiber of the conic bundle $\omega $ . Therefore, we see that $\omega (\ell )\in \Delta _{\mathbb {P}^1\times \mathbb {P}^1}$ . This implies that the conic bundle $\omega $ maps an irreducible component of the conic C to an irreducible component of the curve $\Delta _{\mathbb {P}^1\times \mathbb {P}^1}$ because S is a general surface in the linear system $|H_3|$ that contains the curve C.

If C is singular, an irreducible component of the curve $\Delta _{\mathbb {P}^1\times \mathbb {P}^1}$ is a curve of degree $(1,0)$ or $(0,1)$ , which is impossible [15, §3.8]. Therefore, we see that the conic C is smooth and irreducible, and the curve $\omega (C)\cong C$ is an irreducible component of the discriminant curve $\Delta _{\mathbb {P}^1\times \mathbb {P}^1}$ . Since the conic bundle $\omega $ is standard [15], the surface $\omega ^{-1}(\omega (C))$ is irreducible and nonnormal, which easily implies that the conic C is contained in its singular locus.

Choosing appropriate coordinates on $\mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}\times \mathbb {P}^2_{x,y,z}$ , we may assume that $\pi _3(C)=[0:0:1]$ , the conic C is given by $x=y=sv-tu=0$ , $([0:1],[0:1])$ is a smooth point of the curve $\Delta _{\mathbb {P}^1\times \mathbb {P}^1}$ , and the fiber $\omega ^{-1}([0:1],[0:1])$ is given by $s=u=xy=0$ . Then X is given by

$$ \begin{align*} &(a_1su+b_1sv+c_1 tu)x^2+(a_2su+b_2sv+c_2tu+tv)xy+\\ & \quad +b_4(sv-tu)xz+(a_3su+b_3sv+c_3tu)y^2+b_5(sv-tu)yz+(sv-tu)z^2=0 \end{align*} $$

for some numbers $a_1$ , $a_2$ , $a_3$ , $b_1$ , $b_2$ , $b_3$ , $b_4$ , $b_5$ , $c_1$ , $c_2$ , $c_3$ . One can check that $\Delta _{\mathbb {P}^1\times \mathbb {P}^1}$ indeed splits as a union of the curve $\omega (C)$ and the curve in $\mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}$ of degree $(2,2)$ that is given by

$$ \begin{align*} &a_1b_5^2stu^2-a_1b_5^2s^2uv+a_2b_4b_5s^2uv-a_2b_4b_5stu^2-a_3b_4^2s^2uv+a_3b_4^2stu^2-b_1b_5^2s^2v^2+\\& \quad +b_1b_5^2stuv+b_2b_4b_5s^2v^2-b_2b_4b_5stuv-b_3b_4^2s^2v^2+b_3b_4^2stuv-b_4^2c_3stuv+b_4^2c_3t^2u^2+\\& \quad +b_4b_5c_2stuv-b_4b_5c_2t^2u^2-b_5^2c_1stuv+b_5^2c_1t^2u^2+4a_1a_3s^2u^2+4a_1b_3s^2uv+4a_1c_3stu^2-\\& \quad -a_2^2s^2u^2-2a_2b_2s^2uv-2a_2c_2stu^2+4a_3b_1s^2uv+4a_3c_1stu^2+ 4b_1b_3s^2v^2+4b_1c_3stuv-\\& \quad -b_2^2s^2v^2-2b_2c_2stuv+4b_3c_1stuv+b_4b_5stv^2-b_4b_5t^2uv+4c_1c_3t^2u^2-c_2^2t^2u^2-2a_2stuv-\\& \quad -2b_2stv^2-2c_2t^2uv-t^2v^2=0. \end{align*} $$

The surface S is cut out on X by the equation $y=\lambda x$ , where $\lambda $ is a general complex number. Then the double cover $\beta \colon \overline {S}\to \mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}$ is branched over a singular curve of degree $(2,2)$ , which splits as a union of the curve $\omega (C)$ and the curve in $\mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}$ of degree $(1,1)$ that is given by

$$ \begin{align*} &\lambda^2 b_5^2tu-\lambda^2b_5^2sv+4\lambda^2a_3su+4\lambda^2b_3sv-2b_4\lambda b_5sv+2\lambda b_4b_5tu+\\& \quad +4\lambda^2c_3tu+4\lambda a_2su+4\lambda b_2sv-b_4^2sv+b_4^2tu+4\lambda c_2tu+4a_1su+4b_1sv+4c_1tu+4\lambda tv=0. \end{align*} $$

Since $\lambda $ is general and X is smooth, these two curves intersect transversally by two points, which implies the remaining assertions of the lemma.

Note that the case ( $\mathrm {2}$ ) in Lemma 6 indeed can happen. For instance, if X is given by

$$ \begin{align*}(sv+tu)x^2+(su-sv+tv)xy+(5sv-5tu)zx+3suy^2+(sv-tu)zy+(sv-tu)z^2=0, \end{align*} $$

then X is smooth, and general surface in $|H_3|$ that contains the curve $\pi _3^{-1}([0:0:1])$ is a smooth weak del Pezzo surface, which is not a quartic del Pezzo surface.

Lemma 7. Let C be a smooth fiber of the morphism $\omega $ , and let S be a general surface in $|H_1+H_2|$ that contains the curve C. Then S is a smooth del Pezzo surface of degree $2$ , and $-K_S\sim H_3\vert _{S}$ .

Proof. Left to the reader.

3 Applications of Abban–Zhuang theory

Let us use notations and assumptions of §2. Let $f\colon \widetilde {X}\to X$ be a birational map such that $\widetilde {X}$ is a normal threefold, and let $\mathbf {F}$ be a prime divisor in $\widetilde {X}$ . Then, to prove that X is K-stable, it is enough to show that $\beta (\mathbf {F})=A_X(\mathbf {F})-S_X(\mathbf {F})>0$ , where $A_X(\mathbf {F})=1+\mathrm {ord}_{\mathbf {F}}(K_{\widetilde {X}}/K_X)$ and

$$ \begin{align*}S_X(\mathbf{F})=\frac{1}{-K_X^3}\int_{0}^{\infty}\mathrm{vol}\big(f^*(-K_X)-u\mathbf{F}\big)du. \end{align*} $$

This follows from the valuative criterion for K-stability [11], [13].

Let $\mathfrak {C}$ be the center of the divisor $\mathbf {F}$ on the threefold X. By [10, Th. 10.1], we have

$$ \begin{align*}S_X(S)=\frac{1}{-K_X^3}\int_{0}^{\infty}\mathrm{vol}\big(-K_X-uS\big)du<1 \end{align*} $$

for every surface $S\subset X$ . Hence, if $\mathfrak {C}$ is a surface, then $\beta (\mathbf {F})>0$ . Thus, to show that X is K-stable, we may assume that $\mathfrak {C}$ is either a curve or a point. If $\mathfrak {C}$ is a curve, then [3, Cor. 1.7.26] gives the following corollary.

Corollary 8. Suppose that $\beta (\mathbf {F})\leqslant 0$ and that $\mathfrak {C}$ is a curve. Let S be an irreducible normal surface in the threefold X that contains $\mathfrak {C}$ . Set

$$ \begin{align*} S\big(W^S_{\bullet,\bullet};\mathfrak{C}\big)&=\frac{3}{(-K_X)^3}\int_0^\tau\big(P(u)^{2}\cdot S\big)\cdot\mathrm{ord}_{\mathfrak{C}}\big(N(u)\big\vert_{S}\big)du+\\ & \quad+\frac{3}{(-K_X)^3}\int_0^\tau\int_0^\infty \mathrm{vol}\big(P(u)\big\vert_{S}-v\mathfrak{C}\big)dvdu, \end{align*} $$

where $\tau $ is the largest rational number u such that $-K_X-uS$ is pseudoeffective, $P(u)$ is the positive part of the Zariski decomposition of $-K_X-uS$ , and $N(u)$ is its negative part. Then $S(W^S_{\bullet ,\bullet };\mathfrak {C})>1$ .

Let P be a point in $\mathfrak {C}$ . Then

$$ \begin{align*}\frac{A_X(\mathbf{F})}{S_X(\mathbf{F})}\geqslant\delta_P(X)=\inf_{\substack{E/X\\ P\in C_X(E)}}\frac{A_{X}(E)}{S_X(E)}, \end{align*} $$

where the infimum is taken over all prime divisors E over X whose centers on X that contain P. Therefore, to prove that the Fano threefold X is K-stable, it is enough to show that $\delta _P(X)>1$ . On the other hand, we can estimate $\delta _P(X)$ by using [1, Th. 3.3] and [3, Cor. 1.7.30]. Namely, let S be an irreducible surface in X with Du Val singularities such that $P\in S$ . Set

$$ \begin{align*}\tau=\mathrm{sup}\Big\{u\in\mathbb{Q}_{\geqslant 0}\ \big\vert\ \text{the divisor }-K_X-uS\text{ is pseudoeffective}\Big\}. \end{align*} $$

For $u\in [0,\tau ]$ , let $P(u)$ be the positive part of the Zariski decomposition of the divisor $-K_X-uS$ , and let $N(u)$ be its negative part. Then [1, Th. 3.3] and [3, Cor. 1.7.30] give

(3.1) $$ \begin{align} \delta_P(X)\geqslant\mathrm{min}\Bigg\{\frac{1}{S_X(S)},\delta_{P}\big(S;W^S_{\bullet,\bullet}\big)\Bigg\} \end{align} $$

for

$$ \begin{align*}\delta_{P}\big(S;W^S_{\bullet,\bullet}\big)=\inf_{\substack{F/S,\\ P\subseteq C_S(F)}}\frac{A_S(F)}{S(W^S_{\bullet,\bullet};F)}, \end{align*} $$

where

$$ \begin{align*}S\big(W^S_{\bullet,\bullet}; F\big)=\frac{3}{-K_X^3}\kern-1.3pt\int_0^\tau\!\kern-1.2pt\big(P(u)^{2}\cdot S\big)\cdot\mathrm{ord}_{F}\big(N(u)\big\vert_{S}\big)du+\frac{3}{-K_X^3}\!\int_{0}^{\tau}\!\!\int_0^\infty \!\mathrm{vol}\big(P(u)\big\vert_{S}-vF\big)dvdu, \end{align*} $$

and now the infimum is taken over all prime divisors F over S whose centers on S that contain P. Let us show how to apply (3.1) in some cases. Recall that $S_X(S)<1$ by [10, Th. 10.1].

Lemma 9. Let C be the fiber of the conic bundle $\pi _3$ that contains P, and let S be a general surface in $|H_3|$ that contains C. Suppose that S is a smooth del Pezzo of degree $4$ and that C is smooth. Then $\delta _P(X)>1$ .

Proof. One has $\tau =1$ . Moreover, for $u\in [0,1]$ , we have $N(u)=0$ and $P(u)|_S=-K_S+ (1-u)C$ . Let $L=-K_S+(1-u)C$ . Using Lemma 24 and arguing as in the proof of Lemma 27, we get

$$ \begin{align*} S\big(W^S_{\bullet,\bullet};F\big)&=\frac{1}{6}\int_0^1 4(1+(1-u))S_L(F)du\leqslant \\ & \quad \leqslant A_S(F)\int_0^1 \frac{4}{6}(1+(1-u)) \frac{19+8(1-u)+(1-u)^2}{24}du=\frac{143}{144}A_S(F) \end{align*} $$

for any prime divisor F over S such that $P\in C_S(F)$ . Then (3.1) gives $\delta _P(X)>1$ .

Similarly, we obtain the following result.

Lemma 10. Let S be the surface in $|H_1|$ that contains P. Then

$$ \begin{align*}\delta_P(X)\geqslant\mathrm{min}\Bigg\{\frac{1}{S_X(S)},\frac{2,592\delta_P(S)}{2,560+63\delta_P(S)}\Bigg\} \end{align*} $$

for $\delta _P(S)=\delta _P(S,-K_S)$ , where $\delta _P(S,-K_S)$ is defined in Appendix 1.

Proof. We have $\tau =\frac {3}{2}$ . Moreover, we have

$$ \begin{align*}P(u)=\left\{\begin{aligned} &(1-u)H_1+H_2+H_3,\ \text{if }0\leqslant u\leqslant 1, \\ &(2-u)H_2+(3-2u)H_3,\ \text{if }1\leqslant u\leqslant \frac{3}{2}, \\ \end{aligned} \right. \end{align*} $$

and

$$ \begin{align*}N(u)=\left\{\begin{aligned} &0,\ \text{if }0\leqslant u\leqslant 1, \\ &(u-1)E_2,\ \text{if }1\leqslant u\leqslant \frac{3}{2}.\\ \end{aligned} \right. \end{align*} $$

Note also that $E_2\vert _{S}$ is a smooth genus $3$ curve contained in the smooth locus of the surface S.

Recall that S is a quintic del Pezzo surface with at most Du Val singularities and that the restriction morphism $\pi _2\vert _{S}\colon S\to \mathbb {P}^1_{u,v}$ is a conic bundle. Note that the morphism $\pi _3\vert _{S}\colon S\to \mathbb {P}^2_{x,y,z}$ is birational. Let C be a fiber of the conic bundle $\pi _2\vert _{S}$ , and let L be the preimage in S of a general line in $\mathbb {P}^2_{x,y,z}$ . Then $-K_S\sim C+L$ and

$$ \begin{align*}P(u)\big\vert_{S}\sim_{\mathbb{R}}\left\{\begin{aligned} &C+L,\ \text{if }0\leqslant u\leqslant 1, \\ &(2-u)C+(3-2u)L,\ \text{if }1\leqslant u\leqslant \frac{3}{2}. \\ \end{aligned} \right. \end{align*} $$

Since $2L-C$ is pseudoeffective, the divisor $\frac {7-4u}{3}(-K_S)-(2-u)C-(3-2u)L$ is also pseudoeffective.

Let F be a divisor over S such that $P\in C_S(F)$ . Then it follows from Lemma 27 that

$$ \begin{align*} S\big(W^S_{\bullet,\bullet};F\big)&\leqslant\frac{1}{6}A_S(F)\int_1^{\frac{3}{2}}(u-1)\big(P(u)\big\vert_{S}\big)^2du+\frac{1}{6}\int_{0}^{\frac{3}{2}}\int_0^\infty \mathrm{vol}\big(P(u)\big\vert_{S}-vF\big)dvdu=\\&=\frac{7}{288}A_S(F)+\frac{1}{6}\int_{0}^{1}\int_0^\infty \mathrm{vol}\big(-K_S-vF\big)dvdu+\\&\quad+\frac{1}{6}\int_{1}^{\frac{3}{2}}\int_0^\infty\mathrm{vol}\big((2-u)C+(3-2u)L-vF\big)dvdu\leqslant\\&\leqslant\frac{7}{288}A_S(F)+\frac{1}{6}\int_{0}^{1}5\frac{A_S(F)}{\delta_P(S)}du+\frac{1}{6}\int_{1}^{\frac{3}{2}}\int_0^\infty\mathrm{vol}\Bigg(\frac{7-4u}{3}\big(-K_S\big)-vF\Bigg)dvdu=\\&=\frac{7}{288}A_S(F)+\frac{5}{6\delta_P(S)}A_S(F)+\frac{1}{6}\int_{1}^{\frac{3}{2}}\Bigg(\frac{7-4u}{3}\Bigg)^3\int_0^\infty\mathrm{vol}\big(-K_S-vF\big)dvdu\leqslant\\&\leqslant\frac{7}{288}A_S(F)+\frac{5}{6\delta_P(S)}A_S(F)+\frac{1}{6}\int_{1}^{\frac{3}{2}}\Bigg(\frac{7-4u}{3}\Bigg)^35\frac{A_S(F)}{\delta_P(S)}du=\\&=\frac{7}{288}A_S(F)+\frac{5}{6\delta_P(S)}A_S(F)+\frac{25}{162\delta_P(S)}A_S(F)=\Bigg(\frac{80}{81\delta_P(S)}+\frac{7}{288}\Bigg)A_S(F). \end{align*} $$

Then $\delta _{P}(S;W^S_{\bullet ,\bullet })\geqslant \frac {1}{\frac {80}{81\delta _P(S)}+\frac {7}{288}}=\frac {2,592\delta _P(S)}{2,560+63\delta _P(S)}$ and the required assertion follows from (3.1).

Keeping in mind that $S_X(S)<1$ by [10, Th. 10.1] and the $\delta $ -invariant of the smooth quintic del Pezzo surface is $\frac {15}{13}$ by [3, Lem. 2.11], we obtain the following corollary.

Corollary 11. Let S be the surface in $|H_1|$ that contains P. If S is smooth, then $\delta _P(X)>1$ .

Similarly, using Lemmas 25 and 26 from Appendix 1, we obtain the following corollary.

Corollary 12. Let S be the surface in $|H_1|$ that contains P. Suppose that S has at most singular points of type $\mathbb {A}_1$ and that P is not contained in any line in S that passes through a singular point. Then $\delta _P(X)>1$ .

Alternatively, we can estimate $\delta _P(X)$ using [3, Th. 1.7.30]. Namely, let C be an irreducible smooth curve in S that contains P. Suppose S is smooth at P. Since $S\not \subset \mathrm {Supp}(N(u))$ , we write

$$ \begin{align*}N(u)\big\vert_S=d(u)C+N_S^\prime(u), \end{align*} $$

where $N_S^\prime (u)$ is an effective $\mathbb {R}$ -divisor on S such that $C\not \subset \mathrm {Supp}(N_S^\prime (u))$ , and $d(u)=\mathrm {ord}_C(N(u)\vert _S)$ . Now, for every $u\in [0,\tau ]$ , we define the pseudoeffective threshold $t(u)\in \mathbb {R}_{\geqslant 0}$ as follows:

$$ \begin{align*}t(u)=\inf\Big\{v\in \mathbb R_{\geqslant 0} \ \big|\ \text{the divisor }P(u)\big|_S-vC\text{ is pseudoeffective}\Big\}. \end{align*} $$

For $v\in [0, t(u)]$ , we let $P(u,v)$ be the positive part of the Zariski decomposition of $P(u)|_S-vC$ , and we let $N(u,v)$ be its negative part. As in Corollary 8, we let

$$ \begin{align*} S\big(W^S_{\bullet,\bullet};C\big)&=\frac{3}{(-K_X)^3}\int_0^\tau\big(P(u)^{2}\cdot S\big)\cdot\mathrm{ord}_{C}\big(N(u)\big\vert_{S}\big)du+\\ & \quad + \frac{3}{(-K_X)^3}\int_0^\tau\int_0^\infty \mathrm{vol}\big(P(u)\big\vert_{S}-vC\big)dvdu. \end{align*} $$

Note that $C\not \subset \mathrm {Supp}(N(u,v))$ for every $u\in [0, \tau )$ and that $v\in (0, t(u))$ . Thus, we can let

$$ \begin{align*}F_P\big(W_{\bullet,\bullet,\bullet}^{S,C}\big)=\frac{6}{(-K_X)^3} \int_0^\tau\int_0^{t(u)}\big(P(u,v)\cdot C\big)\cdot \mathrm{ord}_P\big(N_S^\prime(u)\big|_C+N(u,v)\big|_C\big)dvdu. \end{align*} $$

Finally, we let

$$ \begin{align*}S\big(W_{\bullet, \bullet,\bullet}^{S,C};P\big)=\frac{3}{(-K_X)^3}\int_0^\tau\int_0^{t(u)}\big(P(u,v)\cdot C\big)^2dvdu+F_P\big(W_{\bullet,\bullet,\bullet}^{S,C}\big). \end{align*} $$

Then [3, Th. 1.7.30] gives the following corollary.

Corollary 13. One has

(★) $$ \begin{align} \frac{A_X(\mathbf{F})}{S_X(\mathbf{F})}\geqslant\delta_P(X)\geqslant \min\left\{\frac{1}{S(W_{\bullet, \bullet,\bullet}^{S,C}; P)}, \frac{1}{S(W_{\bullet,\bullet}^S;C)},\frac{1}{S_X(S)}\right\}. \end{align} $$

Moreover, if both inequalities in (★) are equalities and $\mathfrak {C}=P$ , then $\delta _P(X)=\frac {1}{S_X(S)}$ .

Let us show how to compute $S(W_{\bullet ,\bullet }^S;C)$ and $S(W_{\bullet , \bullet ,\bullet }^{S,C};P)$ in some cases.

Lemma 14. Suppose that $\omega (P)\not \in \Delta _{\mathbb {P}^1\times \mathbb {P}^1}$ . Let S be a general surface in $|H_1+H_2|$ that contains P, and let C be the fiber of the morphism $\omega $ containing P. Then $S(W_{\bullet ,\bullet }^S;C)=\frac {31}{36}$ and $S(W_{\bullet , \bullet ,\bullet }^{S,C};P)=1$ .

Proof. We have $\tau =1$ . Moreover, for $u\in [0,1]$ , we have $N(u)=0$ and $P(u)|_S=-K_S+2(1-u)C$ . On the other hand, it follows from Lemma 7 that S is a smooth del Pezzo surface of degree $2$ , and the restriction map $\pi _3\vert _{S}\colon S\to \mathbb {P}^2_{x,y,z}$ is a double cover that is ramified over a smooth quartic curve. Therefore, applying the Galois involution of this double cover to C, we obtain another smooth irreducible curve $Z\subset S$ such that $C+Z\sim -2K_S$ , $C^2=Z^2=0$ and $C\cdot Z=4$ , which gives

$$ \begin{align*}P(u)|_S-vC\sim_{\mathbb{R}}\Big(\frac{5}{2}-2u-v\Big)C+\frac{1}{2}Z. \end{align*} $$

Then $P(u)\vert _{S}-vC$ is pseudoeffective $\iff P(u)\vert _{S}-vC$ is nef $\iff v\leqslant \frac {5}{2}-2u$ . Thus, we have

$$ \begin{align*}\mathrm{vol}\big(P(u)\vert_{S}-vC\big)=\big(-K_S+2(1-u)C\big)^2=10-8u-4v \end{align*} $$

and $P(u,v)\cdot C=2$ . Now, integrating, we obtain $S(W_{\bullet ,\bullet }^S;C)=\frac {31}{36}$ and $S(W_{\bullet ,\bullet ,\bullet }^{S,C};P)=1$ .

Lemma 15. Suppose that $P\not \in E_1\cup E_2$ . Let S be a general surface in $|H_3|$ that contains P, and let C be the fiber of the morphism $\pi _3$ containing P. Suppose that S is a smooth del Pezzo surface. Then $S(W_{\bullet ,\bullet }^S;C)=\frac {7}{9}$ and $S(W_{\bullet , \bullet ,\bullet }^{S,C};P)=1$ .

Proof. We have $\tau =1$ . Moreover, for $u\in [0,1]$ , we have $N(u)=0$ and $P(u)|_S=-K_S+(1-u)C$ . Since S is a smooth del Pezzo surface, the restriction map $\omega \vert _{S}\colon S\to \mathbb {P}^1_{s,t}\times \mathbb {P}^1_{u,v}$ is a double cover ramified over a smooth elliptic curve. Therefore, using the Galois involution of this double cover, we get an irreducible curve $Z\subset S$ such that $C+Z\sim -K_S$ , $C^2=Z^2=0$ , and $C\cdot Z=2$ , which gives

$$ \begin{align*}P(u)|_S-vC\sim_{\mathbb{R}}(2-u-v)C+Z. \end{align*} $$

Then $P(u)\vert _{S}-vC$ is pseudoeffective $\iff P(u)\vert _{S}-vC$ is nef $\iff v\leqslant 2-u$ . Thus, we have

$$ \begin{align*}\mathrm{vol}\big(P(u)\vert_{S}-vC\big)=\big(-K_S+(1-u)C\big)^2=8-4u-4v \end{align*} $$

and $P(u,v)\cdot C=2$ . Now, integrating, we obtain $S(W_{\bullet ,\bullet }^S;C)=\frac {7}{9}$ and $S(W_{\bullet ,\bullet ,\bullet }^{S,C};P)=1$ .

Lemma 16. Suppose that $P\not \in E_1\cup E_2$ . Let S be a general surface in $|H_3|$ that contains P, and let C be the fiber of the morphism $\pi _3$ containing P. Suppose S is not a smooth del Pezzo surface. Then $S(W_{\bullet ,\bullet }^S;C)=\frac {8}{9}$ and $S(W_{\bullet , \bullet ,\bullet }^{S,C};P)=\frac {7}{9}$ .

Proof. We have $\tau =1$ . Moreover, for $u\in [0,1]$ , we have $N(u)=0$ and $P(u)|_S=-K_S+(1-u)C$ . It follows from Lemma 6 that S contains two $(-2)$ -curves $\mathbf {e}_1$ and $\mathbf {e}_2$ such that $-K_S\sim 2C+\mathbf {e}_1+\mathbf {e}_2$ . On the surface S, we have $C^2=0$ , $C\cdot \mathbf {e}_1=C\cdot \mathbf {e}_2=1$ , $\mathbf {e}_1^2=\mathbf {e}_2^2=-2$ , and

$$ \begin{align*}P(u)|_S-vC\sim_{\mathbb{R}}(3-u-v)C+\mathbf{e}_1+\mathbf{e}_2. \end{align*} $$

Then $P(u)\vert _{S}-vC$ is pseudoeffective $\iff v\leqslant 3-u$ . Moreover, we have

$$ \begin{align*}P(u,v)&=\left\{\begin{aligned} &(3-u-v)C+\mathbf{e}_1+\mathbf{e}_2,\ \text{if }0\leqslant v\leqslant 1-u, \\ &\frac{3-u-v}{2}\big(2C+\mathbf{e}_1+\mathbf{e}_2\big),\ \text{if }1-u\leqslant v\leqslant 3-u, \\ \end{aligned} \right. \\N(u,v)&=\left\{\begin{aligned} &0,\ \text{if }0\leqslant v\leqslant 1-u, \\ &\frac{u+v-1}{2}(\mathbf{e}_1+\mathbf{e}_2),\ \text{if }1-u\leqslant v\leqslant 3-u, \\ \end{aligned} \right. \\\mathrm{vol}\big(P(u)\vert_{S}-vC\big)&= \left\{\begin{aligned} &8-4u-4v,\ \text{if }0\leqslant v\leqslant 1-u, \\ &(u+v-3)^2,\ \text{if }1-u\leqslant v\leqslant 3-u. \\ \end{aligned} \right.\end{align*} $$

Now, integrating $\mathrm {vol}(P(u)\vert _{S}-vC)$ , we obtain $S(W_{\bullet ,\bullet }^S;C)=\frac {8}{9}$ .

To compute $S(W_{\bullet ,\bullet ,\bullet }^{S,C};P)$ , observe that $F_P(W_{\bullet ,\bullet ,\bullet }^{S,C})=0$ , because $P\not \in \mathbf {e}_1\cup \mathbf {e}_2$ , since S is a general surface in $|H_3|$ that contains C. On the other hand, we have

$$ \begin{align*}P(u,v)\cdot C=\left\{\begin{aligned} &2,\ \text{if }0\leqslant v\leqslant 1-u, \\ &3-u-v,\ \text{if }1-u\leqslant v\leqslant 3-u. \\ \end{aligned} \right. \end{align*} $$

Hence, integrating $(P(u,v)\cdot C)^2$ , we get $S(W_{\bullet ,\bullet ,\bullet }^{S,C};P)=\frac {7}{9}$ as required.

Lemma 17. Suppose $P\in (E_1\cup E_2)\setminus (E_1\cap E_2)$ . Let S be a general surface in $|H_3|$ that contains P, and let C be the irreducible component of the fiber of the conic bundle $\pi _3$ containing P such that $P\in C$ . Then $S(W_{\bullet ,\bullet }^S;C)=1$ and $S(W_{\bullet , \bullet ,\bullet }^{S,C};P)\leqslant \frac {31}{36}$ .

Proof. We have $\tau =1$ . For $u\in [0,1]$ , we have $N(u)=0$ and $P(u)|_S\sim _{\mathbb {R}}-K_S+(1-u) (C+C^\prime )$ , where $C^\prime $ is the irreducible curve in S such that $C+C^\prime $ is the fiber of the conic bundle $\pi _3$ that passes through the point P. Since $P\not \in E_1\cap E_2$ , we see that $P\not \in C^\prime $ .

By Lemma 6, the surface S is a smooth del Pezzo surface of degree $4$ , so we can identify it with a complete intersection of two quadrics in $\mathbb {P}^4$ . Then C and $C^\prime $ are lines in S, and S contains four additional lines that intersect C. Denote them by $L_1$ , $L_2$ , $L_3$ , and $L_4$ , and let $Z=L_1+L_2+L_3+L_4$ . Then the intersections of the curves C, $C^\prime $ , and Z on the surface S are given in the table below.

Observe that $-K_S\sim _{\mathbb {Q}}\frac {3}{2}C+\frac {1}{2}C^\prime +\frac {1}{2}Z$ . This gives $P(u)\vert _{S}-vC\sim _{\mathbb {R}}(\frac {5}{2}-u-v)C+ (\frac {3}{2}-u)C^\prime +\frac {1}{2}Z$ , which implies that $P(u)\vert _{S}-vC$ is pseudoeffective $\iff v\leqslant \frac {5}{2}-u$ .

Moreover, we have

$$ \begin{align*}P(u,v)&=\left\{\begin{aligned} &\Big(\frac{5}{2}-u-v\Big)C+\Big(\frac{3}{2}-u\Big)C^\prime+\frac{1}{2}Z,\ \text{if }0\leqslant v\leqslant 1, \\ &\Big(\frac{5}{2}-u-v\Big)(C+C^\prime)+\frac{1}{2}Z,\ \text{if }1\leqslant v\leqslant 2-u, \\ &\Big(\frac{5}{2}-u-v\Big)(C+C^\prime+Z),\ \text{if }2-u\leqslant v\leqslant \frac{5}{2}-u, \\ \end{aligned} \right.\\[4pt]N(u,v)&=\left\{\begin{aligned} &0,\ \text{if }0\leqslant v\leqslant 1, \\ &(v-1)C^\prime,\ \text{if }1\leqslant v\leqslant 2-u, \\ &(v-1)C^\prime+(v+u-2)Z,\ \text{if }2-u\leqslant v\leqslant \frac{5}{2}-u, \\ \end{aligned} \right.\\[4pt]P(u,v)\cdot C&=\left\{\begin{aligned} &1+v,\ \text{if }0\leqslant v\leqslant 1, \\ &2,\ \text{if }1\leqslant v\leqslant 2-u, \\ &10-4u-4v,\ \text{if }2-u\leqslant v\leqslant \frac{5}{2}-u, \end{aligned} \right.\\[4pt]\mathrm{vol}\big(P(u)\vert_{S}-vC\big)&= \left\{\begin{aligned} &8-v^2-4u-2v,\ \text{if }0\leqslant v\leqslant 1, \\ &9-4u-4v,\ \text{if }1\leqslant v\leqslant 2-u, \\ &(5-2u-2v)^2,\ \text{if }2-u\leqslant v\leqslant \frac{5}{2}-u. \\ \end{aligned} \right. \end{align*} $$

Now, integrating $\mathrm {vol}(P(u)\vert _{S}-vC)$ and $(P(u,v)\cdot C)^2$ , we get $S(W_{\bullet ,\bullet }^S;C)=1$ and

$$ \begin{align*} S\big(W_{\bullet, \bullet,\bullet}^{S,C};P\big)=\frac{5}{6}+F_P\big(W_{\bullet,\bullet,\bullet}^{S,C}\big)&=\frac{5}{6}+\frac{1}{3}\int_0^1\int_0^{\frac{5}{2}-u}\big(P(u,v)\cdot C\big)\cdot \mathrm{ord}_P\big(N(u,v)\big|_C\big)dvdu\leqslant\\ &\leqslant\frac{5}{6}+\frac{1}{3}\int_0^1\int_2^{\frac{5}{2}-u}(10-4u-4v)(v+u-2)dvdu=\frac{31}{36}, \end{align*} $$

because $P\not \in C^\prime $ , and the curves Z and C intersect each other transversally.

4 The proof of Main Theorem

Let us use notations and assumptions of §§2 and 3. Recall that $\mathbf {F}$ is a prime divisor over the threefold X and that $\mathfrak {C}$ is its center in X. To prove Main Theorem, we must show that $\beta (\mathbf {F})>0$ .

Lemma 18. Suppose that $\mathfrak {C}$ is a curve. Then $\beta (\mathbf {F})>0$ .

Proof. Suppose that $\beta (\mathbf {F})\leqslant 0$ . Then $\delta _P(X)\leqslant 1$ for every point $P\in \mathfrak {C}$ . Let us seek for a contradiction.

Let $S_1$ be a general surface in the linear system $|H_1|$ . Then $S_1$ is smooth. Hence, if $S_1\cap \mathfrak {C}\ne \varnothing $ , then $\delta _P(X)\leqslant 1$ for every point $P\in S_1\cap \mathfrak {C}$ , which contradicts Corollary 11. We see that $S_1\cdot \mathfrak {C}=0$ . Similarly, we see that $S_2\cdot \mathfrak {C}=0$ for a general surface $S_2\in |H_2|$ . So, we see that $\omega (\mathfrak {C})$ is a point.

Let C be the scheme fiber of the conic bundle $\omega $ over the point $\omega (\mathfrak {C})$ . Then $\mathfrak {C}$ is an irreducible component of the curve C. If the fiber C is smooth, then we $\mathfrak {C}=C$ .

Suppose that C is smooth. If S is a general surface in the linear system $|H_1+H_2|$ that contains $\mathfrak {C}$ , then $S(W_{\bullet ,\bullet }^S;\mathfrak {C})=\frac {31}{36}<1$ by Lemma 14, which contradicts Corollary 8. So, the curve C is singular.

Note that $\pi _3(\mathfrak {C})$ is a line in $\mathbb {P}^2_{x,y,z}$ . On the other hand, the discriminant curve $\Delta _{\mathbb {P}^2}$ is an irreducible smooth quartic curve in $\mathbb {P}^2_{x,y,z}$ . Therefore, in particular, the line $\pi _3(\mathfrak {C})$ is not contained in $\Delta _{\mathbb {P}^2}$ . Now, let P be a general point in $\mathfrak {C}$ , let Z be the fiber of the conic bundle $\pi _3$ that passes through P, and let S be a general surface in $|H_3|$ that contains the curve Z. Then Z and S are both smooth, and it follows from Lemma 6 that S is a del Pezzo of degree $4$ , so that $\delta _P(X)>1$ by Lemma 9.

Hence, to complete the proof of Main Theorem, we may assume that $\mathfrak {C}$ is a point. Set $P=\mathfrak {C}$ . Let $\mathscr {C}$ be the fiber of the conic bundle $\omega $ that contains P.

Lemma 19. Suppose that $P\not \in E_1\cap E_2$ . Then $\beta (\mathbf {F})>0$ .

Proof. Apply Lemmas 15–17 and Corollary 13.

Thus, to complete the proof of Main Theorem, we may assume, in addition, that $P\in E_1\cap E_2$ . Then the conic $\mathscr {C}$ is smooth at P by Lemma 5. In particular, we see that $\mathscr {C}$ is reduced.

Lemma 20. Suppose that $\mathscr {C}$ is smooth. Then $\beta (\mathbf {F})>0$ .

Proof. Apply Lemma 14 and Corollary 13.

To complete the proof of Main Theorem, we may assume that $\mathscr {C}$ is singular. Write $\mathscr {C}=\ell _1+\ell _2$ , where $\ell _1$ and $\ell _2$ are irreducible components of the conic $\mathscr {C}$ . Then $P\ne \ell _1\cap \ell _2$ , since $P\not \in \mathrm {Sing}(\mathscr {C})$ .

Let $S_1$ and $S_2$ be general surfaces in $|H_1|$ and $|H_2|$ that pass through the point P, respectively. Then $\mathscr {C}=S_1\cap S_2$ , and it follows from Corollary 4 that $S_1$ or $S_2$ is smooth along the conic $\mathscr {C}$ . Without loss of generality, we may assume that $S_1$ is smooth along $\mathscr {C}$ . We let $S=S_1$ .

If S is smooth, then $\delta _P(X)>1$ by Corollary 11. Thus, we may assume that S is singular.

Recall that S is a quintic del Pezzo surface and that $\ell _1$ and $\ell _2$ are lines in its anticanonical embedding. The preimages of the lines $\ell _1$ and $\ell _2$ on the minimal resolution of the surface S are $(-1)$ -curves, which do not intersect $(-2)$ -curves. By Lemma 1 and Remark 2, one of the following cases holds:

• ( $\mathbb {A}_1$ ) The surface S has one singular point of type $\mathbb {A}_1$ .

• ( $2\mathbb {A}_1$ ) The surface S has two singular points of type $\mathbb {A}_1$ .

In both cases, the restriction morphism $\pi _3\vert _{S}\colon S\to \mathbb {P}^2_{x,y,z}$ is birational. In ( $\mathbb {A}_1$ )-case, this morphism contracts three disjoint irreducible smooth rational curves $\mathbf {e}_1$ , $\mathbf {e}_2$ , and $\mathbf {e}_3$ such that $E_1\vert _{S}=2\mathbf {e}_1+\mathbf {e}_2+\mathbf {e}_3$ , the curves $\mathbf {e}_1$ , $\mathbf {e}_2$ , and $\mathbf {e}_3$ are sections of the conic bundle $\pi _2\vert _{S}\colon S\to \mathbb {P}^1_{u,v}$ , the curve $\mathbf {e}_1$ passes through the singular point of the surface S, but $\mathbf {e}_2$ and $\mathbf {e}_3$ are contained in the smooth locus of the surface S. In ( $2\mathbb {A}_1$ )-case, the morphism $\pi _3\vert _{S}$ contracts two disjoint curves $\mathbf {e}_1$ and $\mathbf {e}_2$ such that $E_1\big \vert _{S}=2\mathbf {e}_1+2\mathbf {e}_2$ , the curves $\mathbf {e}_1$ and $\mathbf {e}_2$ are sections of the conic bundle $\pi _2\vert _{S}$ , and each curve among $\mathbf {e}_1$ and $\mathbf {e}_2$ contains one singular point of the surface S. In both cases, we may assume that $\ell _1\cap \mathbf {e}_1\ne \varnothing $ .

Let us identify the surface S with its image in $\mathbb {P}^5$ via the anticanonical embedding $S\hookrightarrow \mathbb {P}^5$ . Then $\ell _1$ and $\ell _2$ and the curves contracted by $\pi _3\vert _{S}$ are lines. In ( $\mathbb {A}_1$ )-case, the surface S contains two additional lines $\ell _3$ and $\ell _4$ such that $\ell _3+\ell _4\sim \ell _1+\ell _2$ , the intersection $\ell _3\cap \ell _4$ is the singular point of the surface S, and the intersection graph of the lines $\ell _1$ , $\ell _2$ , $\ell _3$ , $\ell _4$ , $\mathbf {e}_1$ , $\mathbf {e}_2$ , and $\mathbf {e}_3$ is shown here:

In this picture, we denoted by $\bullet $ the singular point of the surface S. Moreover, on the surface S, the intersections of the lines $\ell _1$ , $\ell _2$ , $\ell _3$ , $\ell _4$ , $\mathbf {e}_1$ , $\mathbf {e}_2$ , and $\mathbf {e}_3$ are given in the table below.

Likewise, in ( $2\mathbb {A}_1$ )-case, the surface S contains one additional line $\ell _3$ such that $2\ell _3\sim \ell _1+\ell _2$ , the line $\ell _3$ passes through both singular points of the del Pezzo surface S, and the intersection graph of the lines on the surface S is shown in the following picture:

As above, the singular points of the surface S are denoted by $\bullet $ . The intersections of the lines $\ell _1$ , $\ell _2$ , $\ell _3$ , $\mathbf {e}_1$ , and $\mathbf {e}_2$ on the surface S are given in the table below.

Remark 21. By [7, Lem. 2.9], the lines in S generate the group $\mathrm {Cl}(S)$ and the cone of effective divisors $\mathrm {Eff}(S)$ , and every extremal ray of the Mori cone $\overline {\mathrm {NE}}(S)$ is generated by the class of a line.

In ( $\mathbb {A}_1$ )-case, the point P is one of the points $\mathbf {e}_1\cap \ell _1$ , $\mathbf {e}_2\cap \ell _2$ , or $\mathbf {e}_3\cap \ell _2$ , because $P\in E_1\cap E_2$ . On the other hand, if $P=\mathbf {e}_2\cap \ell _2$ or $P=\mathbf {e}_3\cap \ell _2$ , it follows from Corollary 12 that $\delta _P(X)>1$ . In ( $2\mathbb {A}_1$ )-case, either $P=\mathbf {e}_1\cap \ell _1$ or $P=\mathbf {e}_2\cap \ell _2$ . Therefore, to complete the proof of Main Theorem, we may assume that $P=\mathbf {e}_1\cap \ell _1$ in both cases.

Now, we will apply Corollary 13 to the surface S with $C=\mathbf {e}_1$ at the point P. We have $\tau =\frac {3}{2}$ . As in the proof of Corollary 10, we see that

$$ \begin{align*}P(u)=\left\{\begin{aligned} &(1-u)H_1+H_2+H_3,\ \text{if }0\leqslant u\leqslant 1, \\ &(2-u)H_2+(3-2u)H_3,\ \text{if }1\leqslant u\leqslant \frac{3}{2}, \\ \end{aligned} \right. \end{align*} $$

and

$$ \begin{align*}N(u)=\left\{\begin{aligned} &0,\ \text{if }0\leqslant u\leqslant 1, \\ &(u-1)E_2,\ \text{if }1\leqslant u\leqslant \frac{3}{2}.\\ \end{aligned} \right. \end{align*} $$

Since $H_1\vert _{S}\sim 0$ , $H_2\vert _{S}\sim \ell _1+\ell _2$ , and $H_3\vert _{S}\sim \ell _1+2\mathbf {e}_1$ , we have

$$ \begin{align*}P(u)\big\vert_{S}-v\mathbf{e}_1\sim_{\mathbb{R}}\left\{\begin{aligned} &(2-v)\mathbf{e}_1+2\ell_1+\ell_2,\ \text{if }0\leqslant u\leqslant 1, \\ &(6-4u-v)\mathbf{e}_1+(5-3u)\ell_1+(2-u)\ell_2,\ \text{if }1\leqslant u\leqslant \frac{3}{2}. \\ \end{aligned} \right. \end{align*} $$

Thus, since the intersection form of the curves $\ell _1$ and $\ell _2$ is semi-negative definite, we get

$$ \begin{align*}t(u)=\left\{\begin{aligned} &2\ \text{if }0\leqslant u\leqslant 1, \\ &6-4u\ \text{if }1\leqslant u\leqslant \frac{3}{2}.\\ \end{aligned} \right. \end{align*} $$

Similarly, if $0\leqslant u\leqslant 1$ , then

$$ \begin{align*}P(u,v)&=\left\{\begin{aligned} &(2-v)\mathbf{e}_1+2\ell_1+\ell_2,\ \text{if }0\leqslant v\leqslant 1, \\ &(2-v)\mathbf{e}_1+(3-v)\ell_1+\ell_2,\ \text{if }1\leqslant v\leqslant 2, \\ \end{aligned} \right. \\[4pt]N(u,v)&=\left\{\begin{aligned} &0,\ \text{if }0\leqslant v\leqslant 1, \\ &(v-1)\ell_1,\ \text{if }1\leqslant v\leqslant 2,\\ \end{aligned} \right. \\P(u,v)\cdot\mathbf{e}_1&=\left\{\begin{aligned} &\frac{v+2}{2},\ \text{if }0\leqslant v\leqslant 1, \\ &\frac{4-v}{2},\ \text{if }1\leqslant v\leqslant 2, \\ \end{aligned} \right.\\[4pt]\mathrm{vol}\big(P(u)\big\vert_{S}-v\mathbf{e}_1\big)&=\left\{\begin{aligned} &\frac{10-4v-v^2}{2},\ \text{if }0\leqslant v\leqslant 1, \\ &\frac{(2-v)(6-v)}{2},\ \text{if }1\leqslant v\leqslant 2.\\ \end{aligned} \right. \end{align*} $$

Likewise, if $1\leqslant u\leqslant \frac {3}{2}$ , then

$$ \begin{align*}P(u,v)&=\left\{\begin{aligned} &(6-4u-v)\mathbf{e}_1+(5-3u)\ell_1+(2-u)\ell_2,\ \text{if }0\leqslant v\leqslant 3-2u, \\ &(6-4u-v)\mathbf{e}_1+(8-5u-v)\ell_1+(2-u)\ell_2,\ \text{if }3-2u\leqslant v\leqslant 6-4u, \\ \end{aligned} \right. \\[4pt]N(u,v)&=\left\{\begin{aligned} &0,\ \text{if }0\leqslant v\leqslant 3-2u, \\ &(v+2u-3)\ell_1,\ \text{if }3-2u\leqslant v\leqslant 6-4u,\\ \end{aligned} \right.\\[4pt]P(u,v)\cdot\mathbf{e}_1&=\left\{\begin{aligned} &\frac{4+v-2u}{2},\ \text{if }0\leqslant v\leqslant 3-2u, \\ &\frac{10-6u-v}{2},\ \text{if }3-2u\leqslant v\leqslant 6-4u, \\ \end{aligned} \right. \\[4pt]\mathrm{vol}\big(P(u)\big\vert_{S}-v\mathbf{e}_1\big)&=\left\{\begin{aligned} &\frac{66+24u^2+4uv-v^2-80u-8v}{2},\ \text{if }0\leqslant v\leqslant 3-2u, \\ &\frac{(6-4u-v)(14-8u-v)}{2},\ \text{if }3-2u\leqslant v\leqslant 6-4u.\\ \end{aligned} \right. \end{align*} $$

Integrating, we get $S(W_{\bullet ,\bullet }^S;\mathbf {e}_1)=\frac {137}{144}$ and $S(W_{\bullet , \bullet ,\bullet }^{S,\mathbf {e}_1};P)=\frac {59}{96}+F_P(W_{\bullet ,\bullet ,\bullet }^{S,\mathbf {e}_1})$ . To compute $F_P(W_{\bullet ,\bullet ,\bullet }^{S,\mathbf {e}_1})$ , we let $Z=E_2\vert _{S}$ . Then Z is a smooth curve of genus $3$ such that $\pi (Z)$ is a smooth quartic in $\mathbb {P}^2_{x,y,z}$ . Moreover, the curve Z is contained in the smooth locus of the surface S, and

$$ \begin{align*}Z\sim\left\{\begin{aligned} &4\mathbf{e}_1+\ell_3+\ell_4+2\ell_1\ \text{in (}\mathbb{A}_1\text{)-case}, \\ &2\ell_1+2\ell_2+2\mathbf{e}_1+2\mathbf{e}_2\ \text{in (}2\mathbb{A}_1\text{)-case}. \\ \end{aligned} \right. \end{align*} $$

In particular, we have $Z\cdot \mathbf {e}_1=1$ . Since $\mathbf {e}_1\not \subset Z$ , we have

$$ \begin{align*}N_S^\prime(u)=\left\{\begin{aligned} &0,\ \text{if }0\leqslant u\leqslant 1, \\ &(u-1)Z,\ \text{if }1\leqslant u\leqslant \frac{3}{2}.\\ \end{aligned} \right. \end{align*} $$

Note that $P\in Z$ , because $P\in E_1\cap E_2$ . Thus, since $\mathbf {e}_1\cdot Z=1$ and $\mathbf {e}_1\cdot \ell _1=1$ , we have

$$ \begin{align*} F_P\big(W_{\bullet,\bullet,\bullet}^{S,\mathbf{e}_1}\big)&=\frac{1}{3}\int_1^{\frac{3}{2}}\!\int_0^{6-4u}\!\big(P(u,v)\cdot \mathbf{e}_1\big)(u-1)dvdu+\frac{1}{3}\int_0^{\frac{3}{2}}\!\int_0^{t(u)}\!\big(P(u,v)\cdot \mathbf{e}_1\big)\big(N(u,v)\cdot \mathbf{e}_1\big)dvdu=\\&=\frac{1}{3}\int_1^{\frac{3}{2}}\int_0^{3-2u}\frac{(4+v-2u)(u-1)}{2}dvdu+\frac{1}{3}\int_1^{\frac{3}{2}}\!\int_{3-2u}^{6-4u}\frac{(10-6u-v)(u-1)}{2}dvdu+\\& \quad +\frac{1}{3}\int_0^{1}\int_1^{2}\frac{(4-v)(v-1)}{2}dvdu+\frac{1}{3}\int_1^{\frac{3}{2}}\!\int_{3-2u}^{6-4u}\frac{(10-6u-v)(v+2u-3)}{2}dvdu=\frac{71}{288}, \end{align*} $$

so that $S(W_{\bullet , \bullet ,\bullet }^{S,\mathbf {e}_1};P)=\frac {31}{36}$ . Now, applying Corollary 13, we get $\delta _P(X)>1$ , because $S_X(S)<1$ . Therefore, we see that $\beta (\mathbf {F})>0$ . By [11], [13], this completes the proof of Main Theorem.

Remark 22. Instead of using Corollary 13, we can finish the proof of Main Theorem as follows. Let F be a divisor over S such that $P\in C_S(F)$ , and let $\mathcal {C}$ be a fiber of the conic bundle $\pi _2\vert _{S}$ . Then, arguing as in the proof of Corollary 10, we get

$$ \begin{align*}S\big(W^S_{\bullet,\bullet};F\big)\leqslant \Bigg(\frac{7}{288}+\frac{5}{6\delta_P(S)}\Bigg)A_S(F)+\frac{1}{6}\int_{1}^{\frac{3}{2}}\int_0^\infty\mathrm{vol}\big((2-u)\mathcal{C}+(3-2u)H_3\big\vert_{S}-vF\big)dvdu. \end{align*} $$

But $\delta _P(S)=1$ by Lemmas 25 and 26, since $P=\mathbf {e}_1\cap \ell _1$ . Thus, we have

(♡) $$ \begin{align} S\big(W^S_{\bullet,\bullet};F\big)&\leqslant\frac{247}{288}A_S(F)+\frac{1}{6}\int_{1}^{\frac{3}{2}}\int_0^\infty\mathrm{vol}\big((2-u)\mathcal{C}+(3-2u)H_3\big\vert_{S}-vF\big)dvdu=\\&=\frac{247}{288}A_S(F)+\frac{1}{6}\int_{1}^{\frac{3}{2}}(3-2u)^3\int_0^\infty\mathrm{vol}\Bigg(\frac{2-u}{3-2u}\mathcal{C}+H_3\big\vert_{S}-vF\Bigg)dvdu=\nonumber\\&=\frac{247}{288}A_S(F)+\frac{1}{6}\int_{1}^{\frac{3}{2}}(3-2u)^3\int_0^\infty\mathrm{vol}\Bigg(-K_S+\frac{u-1}{3-2u}\mathcal{C}-vF\Bigg)dvdu.\nonumber \end{align} $$

Set $L=-K_S+t\mathcal {C}$ for $t\in \mathbb {R}_{\geqslant 0}$ . Then L is ample and $L^2=5+4t$ . Define $\delta _P(S,L)$ as in Appendix 1. Then, applying [3, Cor. 1.7.24] to the flag $P\in \mathbf {e}_1\subset S$ , we get

$$ \begin{align*}\delta_P(S,L)\geqslant \left\{\begin{aligned} &1,\ \text{if }0\leqslant t\leqslant \frac{-3+\sqrt{21}}{6}, \\ &\frac{15+12t}{6t^2+18t+13},\ \text{if }\frac{-3+\sqrt{21}}{6}\leqslant t.\\ \end{aligned} \right. \end{align*} $$

The proof of this inequality is very similar to our computations of $S(W_{\bullet ,\bullet }^S;\mathbf {e}_1)$ and $S(W_{\bullet , \bullet ,\bullet }^{S,\mathbf {e}_1};P)$ , so that we omit the details. Now, we let $t=\frac {u-1}{3-2u}$ . Then $t\geqslant \frac {-3+\sqrt {21}}{6}\iff u\geqslant \frac {3}{2}(1-\frac {1}{\sqrt {21}})$ , so

$$ \begin{align*} &\frac{1}{6}\int_{1}^{\frac{3}{2}}(3-2u)^3\int_0^\infty\mathrm{vol}\big(-K_S+t\mathcal{C}-vF\big)dvdu=\\& \quad =\frac{1}{6}\int_{1}^{\frac{3}{2}}(3-2u)^3(5+4t)S_{L}(F)du\leqslant \frac{1}{6}\int_{1}^{\frac{3}{2}(1-\frac{1}{\sqrt{21}})}(3-2u)^3(5+4t)A_S(F)du+\\& \qquad +\frac{1}{6}\int_{\frac{3}{2}(1-\frac{1}{\sqrt{21}})}^{\frac{3}{2}}(3-2u)^3(5+4t)\frac{15+12t}{6t^2+18t+13}A_{S}(F)du=\frac{247}{2,016}A_{S}(F). \end{align*} $$

Now, using (♡), we get $S(W^S_{\bullet ,\bullet };F)\leqslant \frac {247}{288}A_S(F)+\frac {247}{2,016}A_{S}(F)=\frac {247}{252}A_S(F)$ . Then $\delta _{P}(S;W^S_{\bullet ,\bullet })\geqslant \frac {252}{247}$ , so that $\delta _P(X)>1$ by (3.1), since $S_X(S)<1$ by [10, Th. 10.1].

Appendix A $\delta $ -invariants of del Pezzo surfaces

In this appendix, we present three rather sporadic results about $\delta $ -invariants of del Pezzo surfaces with at most du Val singularities, which are used in the proof of Main Theorem.

Let S be a del Pezzo surface that has at most du Val singularities, let L be an ample $\mathbb {R}$ -divisor on the surface S, and let P be a point in S. Set

$$ \begin{align*}\delta_P(S,L)=\inf_{\substack{F/S\\ P\in C_S(F)}}\frac{A_{S}(F)}{S_{L}(F)}, \end{align*} $$

where infimum is taken over all prime divisors F over S such that $P\in C_S(F)$ , and

$$ \begin{align*}S_{L}(F)=\frac{1}{L^2}\int_0^\infty \mathrm{vol}\big(L-uF\big)du. \end{align*} $$

Example 23. Suppose that S is a smooth cubic surface in $\mathbb {P}^3$ and that $L=-K_S$ . Let T be the hyperplane section of the cubic surface S that is singular at P. Then it follows from [1, Th. 4.6] that

$$ \begin{align*}\delta_P(S,L)=\left\{\begin{aligned} &\frac{3}{2},\ \text{if }T\text{ is a union of three lines such that all of them contains }P,\\ &\frac{27}{17},\ \text{if }T\text{ is a~union of a~line and a~conic that are tangent at }P,\\ &\frac{5}{3},\ \text{if }T\text{ is an irreducible cuspidal cubic curve},\\ &\frac{18}{11},\ \text{if }T\text{ is a union of three lines such that only two of them contain }P,\\ &\frac{9}{25-8\sqrt{6}},\ \text{if }T\text{ is a~union of a~line and a~conic that intersect transversally at }P,\\ &\frac{12}{7},\ \text{if }T\text{ is an irreducible nodal cubic curve}. \end{aligned} \right. \end{align*} $$

It would be nice to find an explicit formula for $\delta _P(S,L)$ in all possible cases. But this problem seems to be very difficult. So, we will only estimate $\delta _P(S,L)$ in three cases when $K_S^2\in \{4,5\}$ .

Suppose that $4\leqslant K_S^2\leqslant 5$ . Let us identify S with its image in the anticanonical embedding.

Lemma 24. Suppose that S is smooth and $K_S^2=4$ . Let C be a possibly reducible conic in S that passes through P, and let $L=-K_S+tC$ for $t\in \mathbb {R}_{\geqslant 0}$ . If the conic C is smooth, then

(♣) $$ \begin{align} \delta_P(S,L)\geqslant \begin{cases} &\frac{24}{19+8t+t^2},\ \text{if }0\leqslant t\leqslant 1, \\ &\frac{6(1+t)}{5+6t+3t^2},\ \text{if }t\geqslant 1. \end{cases} \end{align} $$

Similarly, if C is a reducible conic, then

(♠) $$ \begin{align} \delta_L(S,L)\geqslant \frac{24(1+t)}{19+30t+12t^2}. \end{align} $$

Proof. The proof of this lemma is similar to the proof of [3, Lem. 2.12]. Namely, as in that proof, we will apply [3, Th. 1.7.1], [3, Cor. 1.7.12], and [3, Cor. 1.7.25] to get (♣) and (♠). Let us use notations introduced in [3, Sect. 1] applied to S polarized by the ample divisor L.

First, we suppose that P is not contained in any line in S. In particular, the conic C is smooth. Let $\sigma \colon \widetilde {S}\to S$ be the blowup of the point P, let E be the exceptional curve of the blowup $\sigma $ , and let $\widetilde {C}$ be the proper transform on $\widetilde {S}$ of the conic C. Then $\widetilde {S}$ is a smooth cubic surface in $\mathbb {P}^3$ , and there exists a unique line $\mathbf {l}\subset \widetilde {S}$ such that $-K_{\widetilde {S}}\sim \widetilde {C}+E+\mathbf {l}$ . Take $u\in \mathbb {R}_{\geqslant 0}$ . Then

$$ \begin{align*}\sigma^*(L)-uE\sim_{\mathbb{R}}(1+t)\widetilde{C}+(2+t-u)E+\mathbf{l}, \end{align*} $$

which implies that $\sigma ^*(L)-uE$ is pseudoeffective $\iff u\leqslant 2+t$ . Similarly, we see that

$$ \begin{align*}\mathscr{P}(u)&\sim_{\mathbb{R}}\left\{\begin{aligned} &(1+t)\widetilde{C}+(2+t-u)E+\mathbf{l},\ \text{if }0\leqslant u\leqslant 2, \\ &(3+t-u)\widetilde{C}+(2+t-u)E+\mathbf{l},\ \text{if }2\leqslant u\leqslant2+t, \\ \end{aligned} \right.\\[8pt]\mathscr{N}(u)&=\left\{\begin{aligned} &0,\ \text{if }0\leqslant u\leqslant 2, \\ &(u-2)\widetilde{C},\ \text{if }2\leqslant u\leqslant 2+t,\\ \end{aligned} \right.\\[8pt]\mathscr{P}(u)\cdot E&=\left\{\begin{aligned} &u,\ \text{if }0\leqslant u\leqslant 2, \\ &2,\ \text{if }2\leqslant u\leqslant2+t, \\ \end{aligned} \right.\\[8pt]\mathrm{vol}\big(\sigma^*(L)-uE\big)&=\left\{\begin{aligned} &4+4t-u^2,\ \text{if }0\leqslant u\leqslant 2, \\ &4(2+t-u),\ \text{if }2\leqslant u\leqslant2+t, \\ \end{aligned} \right. \end{align*} $$

where we denote by $\mathscr {P}(u)$ the positive part of the Zariski decomposition of the divisor $\sigma ^*(L)-uE$ , and we denote by $\mathscr {N}(u)$ its negative part. This gives

$$ \begin{align*}S_L(E)=\frac{8+12t+3t^2}{6(1+t)}. \end{align*} $$

Moreover, applying [3, Cor. 1.7.25], we obtain

$$ \begin{align*}S(W^E_{\bullet,\bullet};Q)\leqslant\frac{4+6t+3t^2}{6(1+t)} \end{align*} $$

for every point $Q\in E$ . Note that $A_S(E)=2$ . Thus, it follows from [3, Cor. 1.7.12] that

$$ \begin{align*}\delta_P(S,L)\geqslant\frac{6(1+t)}{4+6t+3t^2}>\frac{24}{19+8t+t^2}. \end{align*} $$

To complete the proof of the lemma, we may assume that S contains a line $\ell $ such that $P\in \ell $ . Then $\ell \cdot C=0$ or $\ell \cdot C=1$ . If $\ell \cdot C=0$ , then $\ell $ must be an irreducible component of the conic C. Let us apply [3, Th. 1.7.1] and [3, Cor. 1.7.25] to the flag $P\in \ell $ to estimate $\delta _P(S,L)$ . Take $u\in \mathbb {R}_{\geqslant 0}$ . Let $P(u)$ be the positive part of the Zariski decomposition of the divisor $L-u\ell $ , and let $N(u)$ be its negative part. We must compute $P(u)$ , $N(u)$ , $P(u)\cdot \ell $ , and $\mathrm {vol}(L-u\ell )$ .

There exists a birational morphism $\pi \colon S\to \mathbb {P}^2$ that blows up five points $O_1,\dots ,O_5\in \mathbb {P}^2$ such that no three of them are collinear. For every $i\in \{1,\ldots ,5\}$ , let $\mathbf {e}_i$ be the $\pi $ -exceptional curve such that $\pi (\mathbf {e}_i)=O_i$ . Similarly, let $\mathbf {l}_{ij}$ be the strict transform of the line in $\mathbb {P}^2$ that contains $O_i$ and $O_j$ , where $1\leqslant i<j\leqslant 5$ . Finally, let B be the strict transform of the conic on $\mathbb {P}^2$ that passes through the points $O_1,\dots ,O_5$ . Then $\mathbf {e}_1,\ldots ,\mathbf {e}_5,\mathbf {l}_{12},\ldots ,\mathbf {l}_{45},B$ are all lines in S, and each extremal ray of the Mori cone $\overline {\mathrm {NE}}(S)$ is generated by a class of one of these $16$ lines.

Suppose that the conic C is irreducible. Then $C\cdot \ell =1$ . In this case, without loss of generality, we may assume that $\ell =\mathbf {e}_1$ and $C\sim \mathbf {l}_{12}+\mathbf {e}_2$ . If $0\leqslant t\leqslant 1$ , then

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &L-u\ell,\ \text{if }0\leqslant u\leqslant 1, \\ &L-u\ell-(u-1)(\mathbf{l}_{12}+\mathbf{l}_{13}+\mathbf{l}_{14}+\mathbf{l}_{15}),\ \text{if }1\leqslant u\leqslant 1+t, \\ &L-u\ell-(u-1)(\mathbf{l}_{12}+\mathbf{l}_{13}+\mathbf{l}_{14}+\mathbf{l}_{15})-(u-t-1)B, \ \text{if }1+t\leqslant u\leqslant\frac{3+t}{2}, \\ \end{aligned} \right. \\[5pt]N(u)&=\left\{\begin{aligned} &0, \ \text{if }0\leqslant u\leqslant 1, \\ &(u-1)(\mathbf{l}_{12}+\mathbf{l}_{13}+\mathbf{l}_{14}+\mathbf{l}_{15}),\ \text{if }1\leqslant u\leqslant 1+t, \\ &(u-1)(\mathbf{l}_{12}+\mathbf{l}_{13}+\mathbf{l}_{14}+\mathbf{l}_{15})+(u-t-1)B, \ \text{if }1+t\leqslant u\leqslant\frac{3+t}{2}, \\ \end{aligned} \right.\\[5pt]P(u)\cdot\ell&=\left\{\begin{aligned} &1+t+u,\ \text{if }0\leqslant u\leqslant 1, \\ &5+t-3u,\ \text{if }1\leqslant u\leqslant 1+t, \\ &6+2t-4u,\ \text{if }1+t\leqslant u\leqslant\frac{3+t}{2}, \\ \end{aligned} \right.\\[5pt]\mathrm{vol}\big(L-u\ell\big)&=\left\{\begin{aligned} &4(1+t)-2u(1+t)-u^2, \ \text{if }0\leqslant u\leqslant 1, \\ &(2-u)(4+2t-3u),\ \text{if }1\leqslant u\leqslant 1+t, \\ &(3+t-2u)^2,\ \text{if }1+t\leqslant u\leqslant\frac{3+t}{2}, \\ \end{aligned} \right.\end{align*} $$

and $L-u\ell $ is not pseudoeffective for $u>\frac {3+t}{2}$ . Similarly, if $t\geqslant 1$ , then

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &L-u\ell,\ \text{if }0\leqslant u\leqslant 1, \\ &L-u\ell-(u-1)(\mathbf{l}_{12}+\mathbf{l}_{13}+\mathbf{l}_{14}+\mathbf{l}_{15}),\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \\[6pt]N(u)&=\left\{\begin{aligned} &0, \ \text{if }0\leqslant u\leqslant 1, \\ &(u-1)(\mathbf{l}_{12}+\mathbf{l}_{13}+\mathbf{l}_{14}+\mathbf{l}_{15}), \ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\\[6pt]P(u)\cdot\ell&=\left\{\begin{aligned} &1+t+u,\ \text{if }0\leqslant u\leqslant 1, \\ &5+t-3u,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\\[6pt]\mathrm{vol}\big(L-u\ell\big)&=\left\{\begin{aligned} &4(1+t)-2u(1+t)-u^2, \ \text{if }0\leqslant u\leqslant 1, \\ &(2-u)(4+2t-3u),\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \end{align*} $$

and $L-u\ell $ is not pseudoeffective for $u>2$ . Then

$$ \begin{align*}S_L\big(\ell\big)= \left\{\begin{aligned} &\frac{17+4t-t^2}{24},\ \text{if }0\leqslant t\leqslant 1, \\ &\frac{2+3t}{3(1+t)},\ \text{if }t\geqslant 1. \\ \end{aligned} \right. \end{align*} $$

Observe that $P\not \in \mathbf {l}_{ij}$ for every $1\leqslant i<j\leqslant 5$ . Thus, if $t\leqslant 1$ , then [3, Cor. 1.7.25] gives

$$ \begin{align*}S(W^{\ell}_{\bullet,\bullet};P)= \left\{\begin{aligned} &\frac{19+8t+t^2}{24},\ \text{if }P\in B, \\ &\frac{9+15t+3t^2+t^3}{12(1+t)},\ \text{if }P\not\in B. \\ \end{aligned} \right. \end{align*} $$

Similarly, if $t\geqslant 1$ , then [3, Cor. 1.7.25] gives

$$ \begin{align*}S\big(W^{\ell}_{\bullet,\bullet};P\big)=\frac{5+6t+3t^2}{6(1+t)}. \end{align*} $$

Now, using [3, Th. 1.7.1], we get (♣).

To complete the proof of the lemma, we may assume that the conic C is reducible. In this case, we let $\ell $ be an irreducible component of the conic C that contains P. Without loss of generality, we may assume that $\ell =\mathbf {e}_1$ and $C=\mathbf {e}_1+B$ . Then

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &L-u\ell,\ \text{if }0\leqslant u\leqslant 1, \\ &L-u\ell-(u-1)B,\ \text{if }1\leqslant u\leqslant 1+t, \\ &L-u\ell-(u-t-1)(\mathbf{l}_{12}+\mathbf{l}_{13}+\mathbf{l}_{14}+\mathbf{l}_{15})-(u-1)B, \ \text{if }1+t\leqslant u\leqslant\frac{3+2t}{2}, \\ \end{aligned} \right. \\[4pt]N(u)&=\left\{\begin{aligned} &0, \ \text{if }0\leqslant u\leqslant 1, \\ &(u-1)B,\ \text{if }1\leqslant u\leqslant 1+t, \\ &(u-t-1)(\mathbf{l}_{12}+\mathbf{l}_{13}+\mathbf{l}_{14}+\mathbf{l}_{15})+(u-1)B, \ \text{if }1+t\leqslant u\leqslant\frac{3+2t}{2}, \\ \end{aligned} \right.\\[4pt]P(u)\cdot\ell&=\left\{\begin{aligned} &1+u,\ \text{if }0\leqslant u\leqslant 1, \\ &2,\ \text{if }1\leqslant u\leqslant 1+t, \\ &6+4t-4u,\ \text{if }1+t\leqslant u\leqslant\frac{3+2t}{2}, \\ \end{aligned} \right. \\[4pt]\mathrm{vol}\big(L-u\ell\big)&=\left\{\begin{aligned} &4(1+t)-2u-u^2,\ \text{if }0\leqslant u\leqslant 1, \\ &5+4t-4u,\ \text{if }1\leqslant u\leqslant 1+t, \\ &(3+2t-2u)^2,\ \text{if }1+t\leqslant u\leqslant\frac{3+2t}{2}, \\ \end{aligned} \right. \end{align*} $$

and the divisor $L-u\ell $ is not pseudoeffective for $u>\frac {3+2t}{2}$ . This gives

$$ \begin{align*}S_L\big(\ell\big)=\frac{17+30t+12t^2}{24(1+t)}. \end{align*} $$

Moreover, using [3, Cor. 1.7.25], we compute

$$ \begin{align*}S\big(W^{\ell}_{\bullet,\bullet};P\big)= \left\{\begin{aligned} &\frac{19+30t+12t^2}{24(1+t)},\ \text{if }P\in B, \\ &\frac{19+24t}{24(1+t)},\ \text{if }P\in\mathbf{l}_{12}\cup\mathbf{l}_{13}\cup\mathbf{l}_{14}\cup\mathbf{l}_{15}, \\ &\frac{3+4t}{4(1+t)},\ \text{otherwise}. \\ \end{aligned} \right. \end{align*} $$

Now, using [3, Th. 1.7.1], we get (♠) as claimed.

In the remaining part of this appendix, we suppose that $K_S^2=5$ , $L=-K_S$ , and S has isolated ordinary double points, that is, singular points of type $\mathbb {A}_1$ . As usual, we set $\delta _P(S)=\delta _P(S,-K_S)$ and

$$ \begin{align*}\delta(S)=\inf_{P\in S}\delta_P(S). \end{align*} $$

Let $\eta \colon \widetilde {S}\to S$ be the minimal resolution of the quintic del Pezzo surface S. Since $-K_{\widetilde {S}}\sim \eta ^*(-K_S)$ , we can estimate the number $\delta _P(S)$ as follows. Let O be a point in the surface $\widetilde {S}$ such that $\eta (O)=P$ , and let C be a smooth irreducible rational curve in $\widetilde {S}$ such that:

• • If $P\in \mathrm {Sing}(S)$ , then C is the $\eta $ -exceptional curve such that $\eta (C)=P$ .

• • If $P\not \in \mathrm {Sing}(S)$ , then C is appropriately chosen curve that contains O.

As usual, we set

$$ \begin{align*}\tau=\mathrm{sup}\Big\{u\in\mathbb{Q}_{\geqslant 0}\ \big\vert\ \text{the divisor }-K_{\widetilde{S}}-uC\text{ is pseudoeffective}\Big\}. \end{align*} $$

For $u\in [0,\tau ]$ , let $P(u)$ be the positive part of the Zariski decomposition of the divisor $-K_{\widetilde {S}}-uC$ , and let $N(u)$ be its negative part. Let

$$ \begin{align*}S_{S}(C)=\frac{1}{K_S^2}\int_{0}^{\infty}\mathrm{vol}\big(-K_{\widetilde{S}}-uC\big)du=\frac{1}{K_S^2}\int_{0}^{\tau}P(u)^2du, \end{align*} $$

and let

$$ \begin{align*}S\big(W^{C}_{\bullet,\bullet},O\big)= \frac{2}{K_S^2}\int_0^\tau\big(P(u)\cdot C\big)\mathrm{ord}_O\big(N(u)\big\vert_{C}\big)du +\frac{1}{K_S^2}\int_0^\tau(P(u)\cdot C)^2du. \end{align*} $$

If $P\not \in \mathrm {Sing}(S)$ , then [3, Th. 1.7.1] and [3, Cor. 1.7.25] give

(⧫) $$ \begin{align} \frac{1}{S_S(C)}\geqslant\delta_P(S)\geqslant\min\left\{\frac{1}{S_S(C)},\frac{1}{S\big(W^{C}_{\bullet,\bullet},O\big)}\right\}. \end{align} $$

Similarly, if $P\in \mathrm {Sing}(S)$ , then [3, Cor. 1.7.12] and [3, Cor. 1.7.25] give

(♢) $$ \begin{align} \frac{1}{S_S(C)}\geqslant\delta_P(S)\geqslant\min\left\{\frac{1}{S_S(C)},\inf_{O\in C}\frac{1}{S\big(W^{C}_{\bullet,\bullet},O\big)}\right\}. \end{align} $$

Lemma 25. Suppose that S has one singular point. Then $\delta (S)=\frac {15}{17}$ , and the following assertions hold:

• • If P is not contained in any line in S that contains the singular point of S, then $\delta _P(S)\geqslant \frac {15}{13}$ .

• • If P is not the singular point of the surface S, but P is contained in a line in S that passes through the singular point of the surface S, then $\delta _P(S)=1$ .

• • If P is the singular point of the surface S, then $\delta _P(S)=\frac {15}{17}$ .

Proof. We let $P_0$ be the singular point of the surface S, and let $\ell _0$ be the $\pi $ -exceptional curve. Then it follows from [8] that there exists a birational morphism $\pi \colon \widetilde {S}\to \mathbb {P}^2$ such that $\pi (\ell _0)$ is a line, the map $\pi $ blows up three points $Q_1$ , $Q_2$ , and $Q_3$ contained in $\pi (\ell _0)$ and another point $Q_0\in \mathbb {P}^2\setminus \pi (\ell _0)$ .

For $i\in \{0,1,2,3\}$ , let $\mathbf {e}_i$ be the $\pi $ -exceptional curve such that $\pi (\mathbf {e}_i)=Q_i$ . For every $i\in \{1,2,3\}$ , let $\ell _i$ be the strict transform of the line in $\mathbb {P}^2$ that passes through $Q_0$ and $Q_i$ . Then $\ell _0$ , $\ell _1$ , $\ell _2$ , $\ell _3$ , $\mathbf {e}_0$ , $\mathbf {e}_1$ , $\mathbf {e}_2$ , and $\mathbf {e}_3$ are the only irreducible curves in the surface $\widetilde {S}$ that have negative self-intersections. Moreover, the intersections of these curves are given in the following table:

Note that $\eta (\ell _1)$ , $\eta (\ell _2)$ , $\eta (\ell _3)$ , $\eta (\mathbf {e}_0)$ , $\eta (\mathbf {e}_1)$ , $\eta (\mathbf {e}_2)$ , and $\eta (\mathbf {e}_3)$ are all lines contained in the surface S. Among them, only the lines $\eta (\mathbf {e}_1)$ , $\eta (\mathbf {e}_2)$ , and $\eta (\mathbf {e}_3)$ pass through the singular point $P_0$ .

For $(a_0,a_1,a_2,a_3,b_0,b_1,b_2,b_3)\in \mathbb {R}^8$ , we write

$$ \begin{align*}[a_0,a_1,a_2,a_3,b_0,b_1,b_2,b_3] := \sum_{i=0}^3 a_i \ell_i + \sum_{i=0}^3 b_i \mathbf{e}_i \in \mathrm{Pic} (\widetilde{S}) \otimes \mathbb{R}. \end{align*} $$

If $P=P_0$ , then $C=\ell _0$ , which implies that $\tau =2$ and

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &[-u, 1, 1, 1, 2, 0, 0, 0],\ \text{if }0\leqslant u\leqslant 1, \\ &[-u, 1, 1, 1, 2, 1-u, 1-u, 1-u],\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \\[6pt]N(u)&=\left\{\begin{aligned} &0,\ \text{if }0\leqslant u\leqslant 1, \\ &(u-1)(\mathbf{e}_1+\mathbf{e}_2+\mathbf{e}_3),\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\\[6pt]P(u)\cdot C&=\left\{\begin{aligned} &2,\ \text{if }0\leqslant u\leqslant 1, \\ &3 -u,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \quad P(u)^2=\left\{\begin{aligned} &5 - 2 u^2,\ \text{if }0\leqslant u\leqslant 1, \\ &(4-u)(2-u),\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \end{align*} $$

which implies that $S_S(C)=\frac {17}{15}$ and $S(W^{C}_{\bullet , \bullet };O)=1$ . Therefore, using (♢), we obtain $\delta _{P_0}(S)=\frac {15}{17}$ .

To proceed, we may assume that $P\ne P_0$ . If $O\in \mathbf {e}_0$ , we let $C=\mathbf {e}_0$ . Then $\tau =2$ , and

$$ \begin{align*}P(u)&=\left\{\begin{aligned} & [0, 1, 1, 1, 2-u, 0, 0, 0],\ \text{if }0\leqslant u\leqslant 1, \\ &[0, 2-u, 2-u, 2-u, 2-u, 0, 0, 0],\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \\[6pt]N(u)&=\left\{\begin{aligned} &0,\ \text{if }0\leqslant u\leqslant 1, \\ &(u-1) (\ell_1 + \ell_2 + \ell_3),\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \\[6pt]P(u)\cdot C&=\left\{\begin{aligned} &1+u,\ \text{if }0\leqslant u\leqslant 1, \\ &4-2u,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \quad P(u)^2=\left\{\begin{aligned} &5-2u-u^2,\ \text{if }0\leqslant u\leqslant 1, \\ &2(2-u)^2,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \end{align*} $$

which implies that $S_S(C)=\frac {13}{15}$ and $S(W^{C}_{\bullet , \bullet };O)\leqslant \frac {13}{15}$ , so that $\delta _P(S)=\frac {15}{13}$ by (⧫).

If $O\in \ell _1$ , we let $C=\ell _1$ . In this case, we have $\tau =2$ , and

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &[0, 1-u, 1, 1, 2, 0, 0, 0],\ \text{if }0\leqslant u\leqslant 1, \\ &[1-u, 1-u, 1, 1, 3-u, 2-2u, 0, 0],\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \\[6pt]N(u)&=\left\{\begin{aligned} &0,\ \text{if }0\leqslant u\leqslant 1, \\ &(u-1)(\ell_0+\mathbf{e}_0+2\mathbf{e}_1),\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\\ [6pt]P(u)\cdot C&=\left\{\begin{aligned} &1+u,\ \text{if }0\leqslant u\leqslant 1, \\ &4-2u,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \quad P(u)^2=\left\{\begin{aligned} &5-2u-u^2,\ \text{if }0\leqslant u\leqslant 1, \\ &2(2-u)^2,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\end{align*} $$

so that $S_S(C)=\frac {13}{15}$ . If $O\in \ell _1\setminus (\mathbf {e}_0\cup \mathbf {e}_1)$ , then $S(W^{C}_{\bullet , \bullet };O)=\frac {11}{15}$ . If $O=\ell _1\cap \mathbf {e}_1$ , then $S(W^{C}_{\bullet , \bullet };O)=1$ . Thus, using (⧫), we see that $\delta _P(S)=\frac {15}{13}$ if $O\in \ell _1\setminus \mathbf {e}_1$ , and $\delta _P(S)\geqslant 1$ if $O=\ell _1\cap \mathbf {e}_1$ .

Similarly, $\delta _P(S)=\frac {15}{13}$ if $O\in \ell _2\setminus \mathbf {e}_2$ or $O\in \ell _3\setminus \mathbf {e}_3$ , and $\delta _P(S)\geqslant 1$ if $O=\ell _2\cap \mathbf {e}_2$ or $O=\ell _3\cap \mathbf {e}_3$ .

If $O\in \mathbf {e}_1$ , we let $C=\mathbf {e}_1$ . In this case, we have $\tau =2$ , and

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &\Big[-\frac{u}{2}, 1, 1, 1, 2, -u, 0, 0\Big],\ \text{if }0\leqslant u\leqslant 1, \\ &\Big[-\frac{u}{2}, 2-u, 1, 1, 2, -u, 0, 0\Big],\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \\[6pt]N(u)&=\left\{\begin{aligned} &\frac{u}{2} \ell_0,\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{u}{2} \ell_0 + (u-1) \ell_1,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\\[6pt]P(u)\cdot C&=\left\{\begin{aligned} &\frac{2+u}{2},\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{4 - u}{2},\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \quad P(u)^2=\left\{\begin{aligned} &5-2u-\frac{u^2}{2},\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{(6-u)(2-u)}{2}, \ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \end{align*} $$

which implies that $S_S(C)=1$ and $S(W^{C}_{\bullet , \bullet };O)\leqslant \frac {13}{15}$ if $O\in \mathbf {e}_1\setminus \ell _0$ , so that $\delta _P(S)=1$ by (⧫).

Likewise, we see that $\delta _P(S)=1$ in the case when $O\in \mathbf {e}_2$ or $O\in \mathbf {e}_3$ . Thus, to complete the proof, we may assume that P is not contained in any line in S.

Now, we let C be the unique curve in the pencil $|\ell _1+\mathbf {e}_1|$ that contains P. By our assumption, the curve C is smooth and irreducible. Then $\tau =2$ , and

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &\Big[-\frac{u}{2}, 1-u, 1, 1, 2, -u, 0, 0\Big],\ \text{if }0\leqslant u\leqslant 1, \\ &\Big[-\frac{u}{2}, 1-u, 1, 1, 3-u, -u, 0, 0\Big],\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \\[6pt]N(u)&=\left\{\begin{aligned} &\frac{u}{2} \ell_0,\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{1}{2} u \ell_0 + (u-1)\mathbf{e}_0,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\\[6pt]P(u)\cdot C&=\left\{\begin{aligned} &\frac{4-u}{2},\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{3(2-u)}{2},\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \quad P(u)^2=\left\{\begin{aligned} &5-4u+\frac{u^2}{2},\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{3(2-u)^2}{2},\ \text{if }1\leqslant u\leqslant 2. \\ \end{aligned} \right.\end{align*} $$

Then $S_S(C)=\frac {11}{15}$ and $S(W^{C}_{\bullet , \bullet };O)=\frac {23}{30}$ . Thus, it follows from (⧫) that $\delta _P (S)\geqslant \frac {30}{23}>\frac {15}{13}$ .

Finally, let us estimate $\delta _P(S)$ in the case when the del Pezzo surface S has two singular points. In this case, the surface S contains a line that passes through both its singular points [8].

Lemma 26. Suppose S has two singular points. Let $\ell $ be the line in S that passes through both singular points of the surface S. Then $\delta (S)=\frac {15}{19}$ . Moreover, the following assertions hold:

• • If P is not contained in any line in S that contains a singular point of S, then $\delta _P(S)\geqslant \frac {15}{13}$ .

• • If P is not contained in the line $\ell $ , but P is contained in a line in S that passes through a singular point of the surface S, then $\delta _P(S)=1$ .

• • If $P\in \ell $ , then $\delta _P(S)=\frac {15}{19}$ .

Proof. Let $\mathbf {e}_1$ and $\mathbf {e}_2$ be $\eta $ -exceptional curves. Then $\widetilde {S}$ contains $(-1)$ -curves $\ell _1$ , $\ell _2$ , $\ell _3$ , $\ell _4$ , and $\ell _5$ such that the intersections of the curves $\ell _1$ , $\ell _2$ , $\ell _3$ , $\ell _4$ , $\ell _5$ , $\mathbf {e}_1$ , and $\mathbf {e}_2$ on $\widetilde {S}$ are given in the following table.

The curves $\eta (\ell _1)$ , $\eta (\ell _2)$ , $\eta (\ell _3)$ , $\eta (\ell _4)$ , and $\eta (\ell _5)$ are the only lines in S. Moreover, we have $\ell =\eta (\ell _1)$ , and $\eta (\ell _1)$ , $\eta (\ell _2)$ , an $\eta (\ell _5)$ are the only lines in S that contain a singular point of the surface S.

As in the proof of Lemma 25, for $(a_1,a_2,a_3,a_4,a_5,b_1, b_2)\in \mathbb {R}^7$ , we write

$$ \begin{align*}[a_1,a_2,a_3,a_4,a_5,b_1, b_2] := \sum_{i=1}^5 a_i \ell_i + \sum_{i=1}^2 b_i \mathbf{e}_i \in \mathrm{Pic} (\widetilde{S}) \otimes \mathbb{R}. \end{align*} $$

If $O\in \ell _1\setminus (\mathbf {e}_1\cup \mathbf {e}_2)$ , we let $C=\ell _1$ . In this case, we have $\tau =3$ , and

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &\Big[1-u, 1, 1, 1, 1, \frac{2-u}{2}, \frac{2-u}{2}\Big], \ \text{if }0\leqslant u\leqslant 2, \\ &[1-u, 3-u, 3-u, 0, 0, 0],\ \text{if }2\leqslant u\leqslant 3, \\ \end{aligned} \right. \\[6pt]N(u)&=\left\{\begin{aligned} &\frac{u}{2} (\mathbf{e}_1 + \mathbf{e}_2),\ \text{if }0\leqslant u\leqslant 2, \\ &(u-2) (\ell_2 + \ell_5) + (u-1)(\mathbf{e}_1 + \mathbf{e}_2), \ \text{if }2\leqslant u\leqslant 3, \\ \end{aligned} \right. \\[6pt]P(u)\cdot C&=\left\{\begin{aligned} &1, \ \text{if }0\leqslant u\leqslant 2, \\ &3-u, \ \text{if }2\leqslant u\leqslant 3, \\ \end{aligned} \right. \quad P(u)^2=\left\{\begin{aligned} &5 - 2u, \ \text{if }0\leqslant u\leqslant 2, \\ &(3-u)^2, \ \text{if }2\leqslant u\leqslant 3, \\ \end{aligned} \right. \end{align*} $$

which implies that $S_S(C)=\frac {19}{15}$ and $S(W^{C}_{\bullet , \bullet };O)\leqslant \frac {17}{15}$ , so that $\delta _P(S)=\frac {15}{19}$ by (⧫).

If $O\in \mathbf {e}_1$ , then $C=\mathbf {e}_1$ . In this case, we have $\tau =2$ , and

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &[1, 1, 1, 1, 1, 1-u, 1],\ \text{if }0\leqslant u\leqslant 1, \\ &[3-2u, 2-u, 1, 1, 1, 1-u, 2-u],\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\\[6pt]N(u)&=\left\{\begin{aligned} &0,\ \text{if }0\leqslant u\leqslant 1, \\ &2(u-1)\ell_1 + (u-1) \ell_2 + (u-1) \mathbf{e}_2,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\\[6pt]P(u)\cdot C&=\left\{\begin{aligned} &2u,\ \text{if }0\leqslant u\leqslant 1, \\ &3-u,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \quad P(u)^2=\left\{\begin{aligned} & 5-2u^2,\ \text{if }0\leqslant u\leqslant 1, \\ &(2-u)(4-u), \ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \end{align*} $$

which implies that $S_S(C)=\frac {17}{15}$ and $S(W^{C}_{\bullet , \bullet };O)\leqslant \frac {19}{15}$ , so that $\delta _P(S)\geqslant \frac {19}{15}$ by (♢).

On the other hand, we already know that $S_S(\ell )=\frac {19}{15}$ , which implies that $\delta _P(S)=\frac {19}{15}$ if $P=\eta (\mathbf {e}_1)$ . Similarly, we see that $\delta _P(S)=\frac {19}{15}$ if $P=\eta (\mathbf {e}_2)$ . Hence, we may assume that $O\not \in \mathbf {e}_1\cup \mathbf {e}_2\cup \ell _1$ .

If $O\in \ell _2$ , we let $C=\ell _2$ . In this case, we have $\tau =2$ , and

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &\Big[1, 1-u, 1, 1, 1, \frac{2-u}{2}, 1\Big],\ \text{if }0\leqslant u\leqslant 1, \\ &\Big[1, 1-u, 2-u, 1, 1, \frac{2-u}{2}, 1\Big],\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \\[6pt]N(u)&=\left\{\begin{aligned} &\frac{u}{2} \mathbf{e}_1,\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{u}{2} \mathbf{e}_1 + (u-1) \ell_3,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\\[6pt]P(u)\cdot C&=\left\{\begin{aligned} &\frac{2+u}{2},\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{4-u}{2},\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \quad P(u)^2=\left\{\begin{aligned} & 5-2u - \frac{u^2}{2},\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{(6-u)(2-u)}{2},\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \end{align*} $$

which implies that $S_S(C)=1$ and $S(W^{C}_{\bullet , \bullet };O)\leqslant \frac {13}{15}$ , so that $\delta _P(S)=1$ by (⧫).

Similarly, we see that $\delta _P(S)=1$ if $O\in \ell _5$ . Hence, if P is contained in a line in S that passes through a singular point of the surface S, then $\delta _P(S)=1$ . Thus, we may assume that $O\not \in \ell _2\cup \ell _2$ .

If $P\in \ell _3$ , we let $C=\ell _3$ . In this case, we have $\tau =2$ , and

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &[1, 1, 1-u, 1, 1, 1, 1],\ \text{if }0\leqslant u\leqslant 1, \\ & [1, 3-2u, 1-u, 2-u, 1, 2-u, 1],\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \\[6pt]N(u)&=\left\{\begin{aligned} &0,\ \text{if }0\leqslant u\leqslant 1, \\ & (u-1) (\ell_4 + 2 \ell_2 + \mathbf{e}_1),\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\\[6pt]P(u)\cdot C&=\left\{\begin{aligned} &1+u,\ \text{if }0\leqslant u\leqslant 1, \\ &4-2u,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \quad P(u)^2=\left\{\begin{aligned} &5-2u - u^2,\ \text{if }0\leqslant u\leqslant 1, \\ &2(2-u)^2,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \end{align*} $$

which implies that $S_S(C)=\frac {13}{15}$ and $S(W^{C}_{\bullet , \bullet };O)\leqslant \frac {13}{15}$ , so that $\delta _P(S)=\frac {15}{13}$ by (⧫).

Similarly, we see that $\delta _P(S)=\frac {15}{13}$ if $O\in \ell _4$ . Therefore, we may also assume that $O\not \in \ell _3\cup \ell _4$ .

Let C be the curve in the pencil $|\ell _2 + \ell _3|$ that contains O. Then C is smooth and irreducible, since O is not contained in the curves $\ell _1$ , $\ell _2$ , $\ell _3$ , $\ell _4$ , $\ell _5$ , $\mathbf {e}_1$ , and $\mathbf {e}_2$ by assumption. Then $\tau =2$ , and

$$ \begin{align*}P(u)&=\left\{\begin{aligned} &\Big[1, 1-u, 1-u, 1, 1, \frac{2-u}{2}, 1\Big],\ \text{if }0\leqslant u\leqslant 1, \\ &\Big[1, 1-u, 1-u, 2-u, 1, \frac{2-u}{2}, 1\Big],\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \\[6pt]N(u)&=\left\{\begin{aligned} &\frac{u}{2} \mathbf{e}_1,\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{u}{2} \mathbf{e}_1 + (u-1) \ell_4,\ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right.\\[6pt]P(u)\cdot C&=\left\{\begin{aligned} &\frac{4-u}{2},\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{3(2-u)}{2}, \ \text{if }1\leqslant u\leqslant 2, \\ \end{aligned} \right. \quad P(u)^2=\left\{\begin{aligned} &5-4u+\frac{u^2}{2},\ \text{if }0\leqslant u\leqslant 1, \\ &\frac{3(2-u)^2}{2},\ \text{if }1\leqslant u\leqslant 2. \\ \end{aligned} \right. \end{align*} $$

This implies that $S_S(C)=\frac {11}{15}$ and $S(W^{C}_{\bullet , \bullet };O)=\frac {23}{30}$ , so that $\delta _P(S)\geqslant \frac {30}{23}>\frac {15}{13}$ by (⧫).

Appendix B Nemuro lemma

Now, let X be any smooth Fano threefold, let $\pi \colon X\to \mathbb {P}^1$ be a fibration into del Pezzo surfaces, let S be a fiber of the morphism $\pi $ such that S is an irreducible reduced normal del Pezzo surface that has at worst du Val singularities, and let P be a point in S. As in §3, set

$$ \begin{align*}\tau=\mathrm{sup}\Big\{u\in\mathbb{Q}_{\geqslant 0}\ \big\vert\ \text{the divisor }-K_X-uS\text{ is pseudoeffective}\Big\}. \end{align*} $$

For $u\in [0,\tau ]$ , let $P(u)$ be the positive part of the Zariski decomposition of the divisor $-K_X-uS$ , and let $N(u)$ be its negative part. Suppose, in addition, that

$$ \begin{align*}N(u)=\sum_{j=1}^l f_j(u) E_j \end{align*} $$

for some irreducible reduced surfaces $E_1,\dots ,E_l$ on the Fano threefold X that are different from S, where each $f_i\colon [0,\tau ]\to \mathbb {R}_{\geqslant 0}$ is some function. For every $j\in \{1,\ldots ,l\}$ , we set $c_j=\mathrm {lct}_{P}(S;E_j|_S)$ . As in Appendix 1, we set $\delta _P(S)=\delta _P(S,-K_S)$ . Define $S(W^S_{\bullet ,\bullet };F)$ and $\delta _{P}(S;W^S_{\bullet ,\bullet })$ as in [3, §1], or define these numbers using the formulas used in (3.1).

Lemma 27. Let F be any prime divisor over S such that $P\in C_S(F)$ . Then

(♢) $$ \begin{align} S\big(W^S_{\bullet,\bullet};F\big)&\leqslant A_S(F)\frac{3}{(-K_X)^3}\int_0^\tau\sum_{j=1}^\tau\frac{f_j(u)}{c_j}\big(P(u)\big|_S\big)^2du+\\& \quad +\frac{3}{(-K_X)^3}\int_0^\tau\int_0^\infty\mathrm{vol}\big(P(u)\big|_S-vF\big)dvdu\leqslant \nonumber\\& \leqslant A_S(F)\Biggl(\frac{3}{(-K_X)^3}\sum_{j=1}^l \int_0^\tau \frac{f_j(u)}{c_j}\big(P(u)\big|_S\big)^2 du+\frac{3}{(-K_X)^3}\frac{\tau(-K_S)^2}{\delta_P(S)}\Biggr).\nonumber \end{align} $$

In particular, we have

$$ \begin{align*} \delta_{P}\big(S;W^S_{\bullet,\bullet}\big)\geqslant\Biggl(\frac{3}{(-K_X)^3}\sum_{j=1}^l \int_0^\tau \frac{f_j(u)}{c_j}\big(P(u)\big|_S\big)^2 du+\frac{3}{(-K_X)^3}\frac{\tau(-K_S)^2}{\delta_P(S)}\Biggr)^{-1}. \end{align*} $$

Proof. Since the log pair $(S, c_j E_j|_S)$ is log canonical at P, we conclude that $\mathrm {ord}_F(E_j|_S)\leqslant \frac {A_S(F)}{c_j}$ . Thus, we get the first inequality in (♢). Moreover, since $P(u)|_S=-K_S-N(u)|_S$ , we have

$$ \begin{align*}\int_0^\tau\int_0^\infty\mathrm{vol}(P(u)|_S-vF\big)dvdu\leqslant\int_0^\tau (-K_S)^2 S_S(F)du =\tau (-K_S)^2 S_S(F)\leqslant A_S(F) \frac{\tau (-K_S)^2}{\delta_P(S)}. \end{align*} $$

Hence, the assertion follows.

Corollary 28. Suppose that $N(u)=0$ for every $u\in [0,\tau ]$ , that is, we have $l=0$ . Then

$$ \begin{align*}\delta_P(S,W^S_{\bullet,\bullet})\geqslant\frac{(-K_X)^3\delta_P(S)}{3\tau(-K_S)^2}. \end{align*} $$

Corollary 29. Suppose that  $l=1$ , $E_1|_S$ is a smooth curve contained in $S\setminus \mathrm {Sing}(S)$ , and

$$ \begin{align*}f_1(u)= \left\{\begin{aligned} &0,\ \text{if }u\in[0,t], \\ &c(u-t),\ \text{if }u\in[t,\tau], \\ \end{aligned} \right. \end{align*} $$

for some $t\in (0,\tau )$ and some $c\in \mathbb {R}_{>0}$ . Then

$$ \begin{align*}\delta_{P}\big(S;W^S_{\bullet,\bullet}\big)\geqslant \Biggl( \frac{3}{(-K_X)^3}\int_t^\tau c(u-t)\big(P(u)\big\vert_S\big)^2du +\frac{3}{(-K_X)^3}\frac{\tau(-K_S)^2}{\delta_P(S)}\Biggr)^{-1}. \end{align*} $$