K-stable divisors in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,2)$

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.

Smooth members of this deformation family are smooth divisors in P 1 × P 1 × P 2 of degree (1,1,2). To be precise, we prove the following result: Main Theorem. Let X be a smooth divisor in P 1 ×P 1 ×P 2 of degree (1,1,2). Then X is K-stable.
We thank the Nemuro city council and Saitama University for excellent working conditions. Cheltsov has been supported by JSPS Invitational Fellowships for Research in Japan (S22040) and by EPSRC Grant Number EP/V054597/1 (The Calabi problem for smooth Fano threefolds). Fujita, Kishimoto and Okada have been supported by JSPS KAKENHI Grants Number 22K03269, 19K03395, JP22H01118, respectively.
We would like to thank an anonymous referee for useful comments.
Throughout this paper, all varieties are assumed to be projective and defined over C. Let X be a divisor in P 1 s,t × P 1 u,v × P 2 x,y,z of tridegree (1,1,2), where ([s : t], [u : v], [x : y : z]) are coordinates on P 1 s,t × P 1 u,v × P 2 x,y,z . Then X is given by the following equation: s t a 11 a 12 a 21 a 22 where each a ij = a ij (x, y, z) is a homogeneous polynomials of degree 2. We can also define X by 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 3 X = 18, and we have the following commutative diagram: x,y,z y y u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ P 2 x,y,z where all maps are induced by natural projections. Note that ω is a (standard) conic bundle whose discriminant curve ∆ P 1 ×P 1 ⊂ P 1 s,t × P 1 u,v is a (possibly singular) curve of degree (3,3) given by Similarly, the map π 3 is a (non-standard) conic bundle whose discriminant curve ∆ P 2 is a smooth plane quartic curve in P 2 x,y,z , which is given by a 11 a 22 = a 12 a 21 . Both maps φ 1 and φ 2 are birational morphisms that blow up the following smooth genus 3 curves: sa 11 + ta 21 = sa 12 + ta 22 = 0 ⊂ P 1 s,t × P 2 x,y,z , ua 11 + va 12 = ua 21 + va 22 = 0 ⊂ P 1 u,v × P 2 x,y,z .
Finally, both morphisms π 1 and π 2 are fibrations into quintic del Pezzo surfaces. 2 Let H 1 = π * 1 (O P 1 (1)), let H 2 = π * 2 (O P 1 (1)), let H 3 = π * 3 (O P 2 (1)), let E 1 and E 2 be the exceptional divisors of the morphisms φ 1 and φ 2 , respectively. Then This gives E 1 + E 2 ∼ 4H 3 , which also follows from E 1 + E 2 = π * 3 (∆ P 2 ). We have In particular, we see that α(X) 2 3 . Note that E 1 ∼ = E 2 ∼ = ∆ P 2 × P 1 . The Mori cone NE(X) is simplicial and is generated by the curves contracted by ω, φ 1 and φ 2 . The cone of effective divisors Eff(X) is generated by the classes of the divisors E 1 , 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 π 3 | S : S → P 2 x,y,z is a birational morphism, and the restriction π 2 | S : S → P 1 u,v is a conic bundle. Moreover, one of the following cases hold: • the surface S is smooth, (A 1 ) the surface S has one singular point of type A 1 , (2A 1 ) the surface S has two singular points of type A 1 , (A 2 ) the surface S has one singular point of type A 2 , (A 3 ) the surface S has one singular point of type A 3 . Furthermore, in each of these five case, the del Pezzo surface S is unique up to an isomorphism.
Remark 2. In the notations and assumptions of Lemma 1, suppose that the surface S is singular, and let ̟ : S → S be its minimal resolution of singularities. Then the dual graph of the (−1)-curves and (−2)-curves on the surface S can be described as follows: (A 1 ) if S has one singular point of type A 1 , then the dual graph is has two singular points of type A 1 , then the dual graph is if S has one singular point of type A 3 , then the dual graph is Here, as in the papers [8,7], we denote a (−1)-curve by •, and we denote a (−2)-curve by •.
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 ∩ S 2 . Then at least one of the surfaces S 1 or S 2 is smooth at P .
Corollary 4. In the notations and assumptions of Lemma 3, suppose the conic S 1 · S 2 is reduced. Then at least one of the surfaces S 1 or S 2 is smooth along S 1 ∩ S 2 .
Lemma 5. Let P be a point in X, let C be the scheme fiber of the conic bundle ω that contains P , and let Z be the scheme fiber of the conic bundle π 3 that contains P . Then C or Z is smooth at P .
Lemma 6. Let C be a fiber of the morphism π 3 , let S be a general surface in |H 3 | that contains C. Then S is smooth, K 2 S = 4 and −K S ∼ (H 1 + H 2 )| S , which implies that −K S is nef and big. Moreover, one of the following three cases holds: (1) the conic C is smooth, −K S is ample, and the restriction ω| S : S → P 1 s,t × P 1 u,v is a double cover branched over a smooth curve of degree (2, 2), (2) the conic C is smooth, the divisor −K S is not ample, the conic ω(C) is an irreducible component of the discriminant curve ∆ P 1 ×P 1 , the conic C is contained in Sing(ω −1 (∆ P 1 ×P 1 )), and the restriction map ω| S : S → P 1 s,t × P 1 u,v fits the following commutative diagram: where α is a birational morphism that contracts two disjoint (−2)-curves, and β is a double cover branched over a singular curve of degree (2, 2), which is a union of the curve ω(C) and another smooth curve of degree (1, 1), which intersect transversally at two distinct points, (3) the conic C is singular, −K S is ample, and the restriction ω| S : S → P 1 s,t × 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 ω| S : where α is a birational morphism that contracts all (−2)-curves in S, and β is a double cover branched over a singular curve of degree (2,2). Let ℓ be a (−2)-curve in S. Then so that ω(ℓ) is a point in P 1 s,t × P 1 u,v . But π 3 (ℓ) is a line in P 2 x,y,z that contains the point π 3 (C). This shows that the curve ℓ is an irreducible component of a singular fiber of the conic bundle ω. Therefore, we see that ω(ℓ) ∈ ∆ P 1 ×P 1 . This implies that the conic bundle ω maps an irreducible component of the conic C to an irreducible component of the curve ∆ P 1 ×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 ∆ P 1 ×P 1 is a curve of degree (1, 0) or (0, 1), which is impossible [16, § 3.8]. Therefore, we see that the conic C is smooth and irreducible, and the curve ω(C) ∼ = C is an irreducible component of the discriminant curve ∆ P 1 ×P 1 . Since the conic bundle ω is standard [16], the surface ω −1 (ω(C)) is irreducible and non-normal, which easily implies that the conic C is contained in its singular locus.
Choosing appropriate coordinates on P 1 s,t × P 1 u,v × P 2 x,y,z , we may assume that π 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 ∆ P 1 ×P 1 , and the fiber ω −1 ([0 : 1], [0 : 1]) is given by s = u = xy = 0. Then X is given by One can check that ∆ P 1 ×P 1 indeed splits as a union of the curve ω(C) and the curve in P 1 s,t × P 1 u,v of degree (2, 2) that is given by The surface S is cut out on X by the equation y = λx, where λ is a general complex number. Then the double cover β : S → P 1 s,t × P 1 u,v is branched over a singular curve of degree (2, 2), which splits as a union of the curve ω(C) and the curve in P 1 s,t × P 1 u,v of degree (1, 1) that is given by sv + 4λ 2 a 3 su + 4λ 2 b 3 sv − 2b 4 λb 5 sv + 2λb 4 b 5 tu+ + 4λ 2 c 3 tu + 4λa 2 su + 4λb 2 sv − b 2 4 sv + b 2 4 tu + 4λc 2 tu + 4a 1 su + 4b 1 sv + 4c 1 tu + 4λtv = 0. Since λ is general and X is smooth, these two curves intersect transversally by two points, which implies the remaining assertions of the lemma.
Lemma 7. Let C be a smooth fiber of the morphism ω, 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 ∼ H 3 | S .
Proof. Left to the reader.

Applications of Abban-Zhuang theory
Let us use notations and assumptions of Section 2. Let f : X → X be a birational map such that X is a normal threefold, and let F be a prime divisor in X. Then, to prove that X is K-stable, it is enough to show that β(F) = A X (F) − S X (F) > 0, where A X (F) = 1 + ord F (K X /K X ) and This follows from the valuative criterion for K-stability [12,14].
Let C be the center of the divisor F on the threefold X. By [11,Theorem 10.1], we have for every surface S ⊂ X. Hence, if C is a surface, then β(F) > 0 . Thus, to show that X is K-stable, we may assume that C is either a curve or a point. If C is a curve, then [3, Corollary 1.7.26] gives Corollary 8. Suppose that β(F) 0 and C is a curve. Let S be an irreducible normal surface in the threefold X that contains C. Set 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 δ P (X) > 1.
On the other hand, we can estimate δ P (X) by using [1, Theorem 3.3] and [3, Corollary 1.7.30]. Namely, let S be an irreducible surface in X with Du Val singularities such that P ∈ S. Set For u ∈ [0, τ ], 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.
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 [11, Theorem 10.1].
Lemma 9. Let C be the fiber of the conic bundle π 3 that contains P , and let S be a general surface in |H 3 | that contains C. Suppose S is a smooth del Pezzo of degree 4, and C is smooth. Then δ P (X) > 1.
Similarly, we obtain the following result: Lemma 10. Let S be the surface in |H 1 | that contain P . Then Note also that E 2 | 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 the restriction morphism π 2 | S : S → P 1 u,v is a conic bundle. Note that the morphism π 3 | S : S → P 2 x,y,z is birational. Let C be a fiber of the conic bundle π 2 | S , and let L be the preimage in S of a general line in P 2 x,y,z . Then −K S ∼ C + L and Let F be a divisor over S such that P ∈ C S (F ). Then it follows from Lemma 27 that 2560+63δ P (S) and the required assertion follows from (3.1).
Keeping in mind that S X (S) < 1 by [11, Theorem 10.1] and the δ-invariant of the smooth quintic del Pezzo surface is 15 13 by [3, Lemma 2.11], we obtain Corollary 11. Let S be the surface in |H 1 | that contain P . If S is smooth, then δ P (X) > 1.
Similarly, using Lemmas 25 and 26 from Appendix A, we obtain Corollary 12. Let S be the surface in |H 1 | that contain P . Suppose that S has at most singular points of type A 1 , and P is not contained in any line in S that passes through a singular point. Then δ P (X) > 1.
Alternatively, we can estimate δ P (X) using [3, Theorem 1.7.30]. Namely, let C be an irreducible smooth curve in S that contains P . Suppose S is smooth at P . Since S ⊂ Supp(N(u)), we write , and d(u) = ord C (N(u)| S ). Now, for every u ∈ [0, τ ], we define the pseudo-effective threshold t(u) ∈ R 0 as follows: For v ∈ [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 Note that C ⊂ Supp(N(u, v)) for every u ∈ [0, τ ) and v ∈ (0, t(u)). Thus, we can let Finally, we let Then [3, Theorem 1.7.30] gives Corollary 13. One has Moreover, if both inequalities in (⋆) are equalities and C = P , then δ P (X) = 1 S X (S) . Let us show how to compute S(W S •,• ; C) and S(W S,C •,•,• ; P ) in some cases. Lemma 14. Suppose that ω(P ) ∈ ∆ P 1 ×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 ω containing P . Then On the other hand, it follows from Lemma 7 that S is a smooth del Pezzo surface of degree 2, and the restriction map π 3 | S : S → 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 ⊂ S such that C + Z ∼ −2K S , C 2 = Z 2 = 0 and C · Z = 4, which gives and P (u, v) · C = 2. Now, integrating, we get S(W S •,• ; C) = 31 36 and S(W S,C •,•,• ; P ) = 1. Lemma 15. Suppose that P ∈ E 1 ∪ E 2 . Let S be a general surface in |H 3 | that contains P , and let C be the fiber of the morphism π 3 containing P . Suppose that S is a smooth del Pezzo surface. Then S(W S •,• ; C) = 7 9 and S(W S,C •,•,• ; P ) = 1. Proof. We have τ = 1. Moreover, for u ∈ [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 ω| S : S → P 1 s,t × 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 ⊂ S such that C + Z ∼ −K S , C 2 = Z 2 = 0, C · Z = 2, which gives and P (u, v) · C = 2. Now, integrating, we obtain S(W S •,• ; C) = 7 9 and S(W S,C •,•,• ; P ) = 1. Lemma 16. Suppose that P ∈ E 1 ∪ E 2 . Let S be a general surface in |H 3 | that contains P , and let C be the fiber of the morphism π 3 containing P . Suppose S is not a smooth del Pezzo surface. Then S(W S •,• ; C) = 8 9 and S(W S,C •,•,• ; P ) = 7 9 . Proof. We have τ = 1. Moreover, for u ∈ [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 e 1 and e 2 such that −K S ∼ 2C + e 1 + e 2 . On the surface S, we have C 2 = 0, C · e 1 = C · e 2 = 1, e 2 1 = e 2 2 = −2, and Then P (u)| S − vC is pseudoeffective ⇐⇒ v 3 − u. Moreover, we have Now, integrating vol(P (u)| S − vC), we obtain S(W S •,• ; C) = 8 9 . To compute S(W S,C •,•,• ; P ), observe that F P (W S,C •,•,• ) = 0, because P ∈ e 1 ∪ e 2 , since S is a general surface in |H 3 | that contains C. On the other hand, we have Hence, integrating (P (u, v) · C) 2 , we get S(W S,C •,•,• ; P ) = 7 9 as required. Lemma 17. Suppose P ∈ (E 1 ∪E 2 ) \ (E 1 ∩E 2 ). Let S be a general surface in |H 3 | that contains P , let C be the irreducible component of the fiber of the conic bundle π 3 containing P such that P ∈ C.
where C ′ is the irreducible curve in S such that C + C ′ is the fiber of the conic bundle π 3 that passes through the point P . Since P ∈ E 1 ∩ E 2 , we see that P ∈ C ′ . 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 P 4 . Then C and C ′ are lines in S, and S contains four additional lines that intersect C. Denote them by L 1 , L 2 , L 3 , L 4 , and let Z = L 1 + L 2 + L 3 + L 4 . Then the intersections of the curves C, C ′ and Z on the surface S are given in the table below.
Now, integrating vol(P (u)| S − vC) and (P (u, v) · C) 2 , we get S(W S •,• ; C) = 1 and because P ∈ C ′ , and the curves Z and C intersect each other transversally.

The proof of Main Theorem
Let us use notations and assumptions of Sections 2 and 3. Recall that F is a prime divisor over the threefold X, and C is its center in X. To prove Main Theorem, we must show that β(F) > 0.
Proof. Suppose β(F) 0. Then δ P (X) 1 for every point P ∈ 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 ∩ C = ∅, then δ P (X) 1 for every point P ∈ S 1 ∩ C, which contradicts Corollary 11. We see that S 1 · C = 0. Similarly, we see that S 2 · C = 0 for a general surface S 2 ∈ |H 2 |. So, we see that ω(C) is a point.
Let C be the scheme fiber of the conic bundle ω over the point ω(C). Then C is an irreducible component of the curve C. If the fiber C is smooth, then we C = C.
Suppose that C is smooth. If S is a general surface in the linear system |H 1 +H 2 | that contains C, then S(W S •,• ; C) = 31 36 < 1 by Lemma 14, which contradicts Corollary 8. So, the curve C is singular. Note that π 3 (C) is a line in P 2 x,y,z . On the other hand, the discriminant curve ∆ P 2 is an irreducible smooth quartic curve in P 2 x,y,z . Therefore, in particular, the line π 3 (C) is not contained in ∆ P 2 . Now, let P be a general point in C, let Z be the fiber of the conic bundle π 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 δ P (X) > 1 by Lemma 9.
Hence, to complete the proof of Main Theorem, we may assume that C is a point. Set P = C. Let C be the fiber of the conic bundle ω that contains P .
Thus, to complete the proof of Main Theorem, we may assume, in addition, that P ∈ E 1 ∩ E 2 . Then the conic C is smooth at P by Lemma 5. In particular, we see that C is reduced.
To complete the proof of Main Theorem, we may assume that C is singular. Write C = ℓ 1 + ℓ 2 , where ℓ 1 and ℓ 2 are irreducible components of the conic C . Then P = ℓ 1 ∩ ℓ 2 , since P ∈ Sing(C ).
Let S 1 and S 2 be general surfaces in |H 1 | and |H 2 | that passes through the point P , respectively. Then C = S 1 ∩ S 2 , and it follows from Corollary 4 that S 1 or S 2 is smooth along the conic C . Without loss of generality, we may assume that S 1 is smooth along C . We let S = S 1 .
If S is smooth, then δ P (X) > 1 by Corollary 11. Thus, we may assume that S is singular.
Recall that S is a quintic del Pezzo surface, and ℓ 1 and ℓ 2 are lines in its anticanonical embedding. The preimages of the lines ℓ 1 and ℓ 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: (A 1 ) the surface S has one singular point of type A 1 , (2A 1 ) the surface S has two singular points of type A 1 .
In both cases, the restriction morphism π 3 | S : S → P 2 x,y,z is birational. In (A 1 )-case, this morphism contracts three disjoint irreducible smooth rational curves e 1 , e 2 , e 3 such that E 1 | S = 2e 1 + e 2 + e 3 , the curves e 1 , e 2 , e 3 are sections of the conic bundle π 2 | S : S → P 1 u,v , the curve e 1 passes through the singular point of the surface S, but e 2 and e 3 are contained in the smooth locus of the surface S. In (2A 1 )-case, the morphism π 3 | S contracts two disjoint curves e 1 and e 2 such that E 1 S = 2e 1 +2e 2 , the curves e 1 and e 2 are sections of the conic bundle π 2 | S , and each curve among e 1 and e 2 contains one singular point of the surface S. In both cases, we may assume that ℓ 1 ∩ e 1 = ∅.
Let us identify the surface S with its image in P 5 via the anticanonical embedding S ֒→ P 5 . Then ℓ 1 and ℓ 2 and the curves contracted by π 3 | S are lines. In (A 1 )-case, the surface S contains two additional lines ℓ 3 and ℓ 4 such that ℓ 3 + ℓ 4 ∼ ℓ 1 + ℓ 2 , the intersection ℓ 3 ∩ ℓ 4 is the singular point of the surface S, and the intersection graph of the lines ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 , e 1 , e 2 , e 3 is shown here: In this picture, we denoted by • the singular point of the surface S. Moreover, on the surface S, the intersections of the lines ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 , e 1 , e 2 , e 3 are given in the table below. Likewise, in (2A 1 )-case, the surface S contains one additional lines ℓ 3 such that 2ℓ 3 ∼ ℓ 1 + ℓ 2 , the line ℓ 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, singular points of the surface S are denote by •. The intersections of the lines ℓ 1 , ℓ 2 , ℓ 3 , e 1 , e 2 on the surface S are given in the table below. In (A 1 )-case, the point P is one of the points e 1 ∩ ℓ 1 , e 2 ∩ ℓ 2 or e 3 ∩ ℓ 2 , because P ∈ E 1 ∩ E 2 . On the other hand, if P = e 2 ∩ ℓ 2 or P = e 3 ∩ ℓ 2 , it follows from Corollary 12 that δ P (X) > 1. In (2A 1 )-case, either P = e 1 ∩ℓ 1 or P = e 2 ∩ℓ 2 . Therefore, to complete the proof of Main Theorem, we may assume that P = e 1 ∩ ℓ 1 in both cases. Now, we will apply Corollary 13 to the surface S with C = e 1 at the point P . We have τ = 3 2 . As in the proof of Corollary 10, we see that and Since Thus, since the intersection form of the curves ℓ 1 and ℓ 2 is semi-negative definite, we get Similarly, if 0 u 1, then Integrating, we get S(W S •,• ; e 1 ) = 137 144 and S(W S,e 1 •,•,• ; P ) = 59 96 + F P (W S,e 1 •,•,• ). To compute F P (W S,e 1 •,•,• ), we let Z = E 2 | S . Then Z is a smooth curve of genus 3 such that π(Z) is a smooth quartic in P 2 x,y,z . Moreover, the curve Z is contained in the smooth locus of the surface S, and In particular, we have Z · e 1 = 1. Since e 1 ⊂ Z, we have 14 Note that P ∈ Z, because P ∈ E 1 ∩ E 2 . Thus, since e 1 · Z = 1 and e 1 · ℓ 1 = 1, we have so that S(W S,e 1 •,•,• ; P ) = 31 36 . Now, applying Corollary 13, we get δ P (X) > 1, because S X (S) < 1. Therefore, we see that β(F) > 0. By [12,14], 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 ∈ C S (F ), and let C be a fiber of the conic bundle π 2 | S . Then, arguing as in the proof of Corollary 10, we get But δ P (S) = 1 by Lemmas 25 and 26, since P = e 1 ∩ ℓ 1 . Thus, we have Set L = −K S +tC for t ∈ R 0 . Then L is ample and L 2 = 5+4t. Define δ P (S, L) as in Appendix A. Then, applying [3, Corollary 1.7.24] to the flag P ∈ e 1 ⊂ S, we get t.
The proof of this inequality is very similar to our computations of S(W S •,• ; e 1 ) and S(W S,e 1 •,•,• ; P ), so that we omit the details. Now, we let t = u−1 3−2u . Then t −3+ Appendix A. δ-invariants of del Pezzo surfaces In this appendix, we present three rather sporadic results about δ-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 R-divisor on the surface S, and let P be a point in S. Set where infimum is taken over all prime divisors F over S such that P ∈ C S (F ), and Example 23. Suppose S is a smooth cubic surface in P 3 , and L = −K S . Let T be the hyperplane section of the cubic surface S that is singular at P . Then it follows from [1, Theorem 4.6] that if T is a union of three lines such that only two of them contain P , if T is a union of a line and a conic that intersect transversally at P , 12 7 if T is an irreducible nodal cubic curve.
It would be nice to find an explicit formula for δ P (S, L) in all possible cases. But this problem seems to be very difficult. So, we will only estimate δ P (S, L) in three cases when K 2 S ∈ {4, 5}. Suppose that 4 K 2 S 5. Let us identify S with its image in the anticanionical embedding.
Lemma 24. Suppose that S is smooth and K 2 S = 4. Let C be a possibly reducible conic in S that passes through P , and let L = −K S + tC for t ∈ R 0 . If the conic C is smooth, then Similarly, if C is a reducible conic, then Proof. The proof of this lemma is similar to the proof of [3, Lemma 2.12]. Namely, as in that proof, we will apply [3, Theorem 1. First, we suppose that P is not contained in any line in S. In particular, the conic C is smooth. Let σ : S → S be the blowup of the point P , let E be the exceptional curve of the blow up σ, and let C be the proper transform on S of the conic C. Then S is a smooth cubic surface in P 3 , and there exists a unique line l ⊂ S such that −K S ∼ C + E + l. Take u ∈ R 0 . Then which implies that σ * (L) − uE is pseudoeffective ⇐⇒ u 2 + t. Similarly, we see that where we denote by P(u) the positive part of the Zariski decomposition of the divisor σ * (L) − uE, and we denote by N (u) its negative part. This gives S L (E) = 8 + 12t + 3t 2 6(1 + t) . To complete the proof of the lemma, we may assume that S contains a line ℓ such that P ∈ ℓ. Then ℓ · C = 0 or ℓ · C = 1. If ℓ · C = 0, then ℓ must be an irreducible component of the conic C. Let us apply [3, Theorem 1.7.1] and [3,Corollary 1.7.25] to the flag P ∈ ℓ to estimate δ P (S, L). Take u ∈ R 0 . Let P (u) be the positive part of the Zariski decomposition of the divisor L − uℓ, and let N(u) be its negative part. We must compute P (u), N(u), P (u) · ℓ and vol(L − uℓ), There exists a birational morphism π : S → P 2 that blows up five points O 1 , . . . , O 5 ∈ P 2 such that no three of them are collinear. For every i ∈ {1, . . . , 5}, let e i be the π-exceptional curve such that π(e i ) = O i . Similarly, let l ij be the strict transform of the line in P 2 that contains O i and O j , where 1 i < j 5. Finally, let B be the strict transform of the conic on P 2 that passes through the points O 1 , . . . , O 5 . Then e 1 , . . . , e 5 , l 12 , . . . , l 45 , B are all lines in S, and each extremal ray of the Mori cone NE(S) is generated by a class of one of these 16 lines.
Suppose that the conic C is irreducible. Then C · ℓ = 1. In this case, without loss of generality, we may assume that ℓ = e 1 and C ∼ l 12 + e 2 . If 0 t 1, then and L − uℓ is not pseudoeffective for u > 3+t 2 . Similarly, if t 1, then and L − uℓ is not pseudoeffective for u > 2. Then Observe that P ∈ l ij for every 1 i < j 5. To complete the proof of the lemma, we may assume that the conic C is reducible. In this case, we let ℓ be an irreducible component of the conic C that contains P . Without loss of generality, 18 we may assume that ℓ = e 1 and C = e 1 + B. Then and the divisor L − uℓ is not pseudoeffective for u > 3+2t 2 . This gives Moreover, using [ In the remaining part of this appendix, we suppose that K 2 S = 5, L = −K S , and S has isolated ordinary double points, i.e. singular points of type A 1 . As usual, we set δ P (S) = δ P (S, −K S ) and Let η : S → S be the minimal resolution of the quintic del Pezzo surface S. Since −K S ∼ η * (−K S ), we can estimate the number δ P (S) as follows. Let O be a point in the surface S such that η(O) = P , and let C be a smooth irreducible rational curve in S such that • if P ∈ Sing(S), then C is the η-exceptional curve such that η(C) = P , • if P ∈ Sing(S), then C is appropriately chosen curve that contains O. As usual, we set τ = sup u ∈ Q 0 the divisor −K S − uC is pseudo-effective .
For u ∈ [0, τ ], let P (u) be the positive part of the Zariski decomposition of the divisor −K S − uC, and let N(u) be its negative part. Let If P ∈ Sing(S), then [ Lemma 25. Suppose S has one singular point. Then δ(S) = 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 δ P (S) 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 δ P (S) = 1. • If P is the singular point of the surface S, then δ P (S) = 15 17 . Proof. We let P 0 be the singular point of the surface S, and let ℓ 0 be the π-exceptional curve. Then it follows from [8] that there exists a birational morphism π : S → P 2 such that π(ℓ 0 ) is a line, the map π blows up three points Q 1 , Q 2 , Q 3 contained in π(ℓ 0 ) and another point Q 0 ∈ P 2 \ π(ℓ 0 ).
Likewise, we see that δ P (S) = 1 in the case when O ∈ e 2 or O ∈ 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 |ℓ 1 + e 1 | that contains P . By our assumption, the curve C is smooth and irreducible. Then τ = 2, and P (u) = Then S S (C) = 11 15 and S(W C •,• ; O) = 23 30 . Thus, it follows from ( ) that δ P (S) 30 23 > 15 13 . Finally, let us estimate δ 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 ℓ be the line in S that passes through both singular points of the surface S. Then δ(S) = 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 δ P (S) 15 13 . • If P is not contained in the line ℓ, but P is contained in a line in S that passes through a singular point of the surface S, then δ P (S) = 1. • If P ∈ ℓ, then δ P (S) = 15 19 .
As in the proof of Lemma 25, for (a 1 , a 2 , a 3 , a 4 , a 5 , b 1 , b 2 ) ∈ R 7 , we write b i e i ∈ Pic( S) ⊗ R.