Delta invariants of smooth cubic surfaces

We prove that $\delta$-invariants of smooth cubic surfaces are at least $\frac{6}{5}$.


Introduction
Let X be a Fano variety with at most Kawamata log terminal singularities. An important problem in algebraic geometry is to find a simple criterion to test the K-stability of the variety X (for notions of stability, we refer the reader to [BHJ15]). The first such criterion is due to Tian, which was generalized by Odaka and Sano.
Two-dimensional Fano varieties are also known as del Pezzo surfaces. In this case, all possible values of α-invariants have been computed in [C08].
Example 1.2. Let S be a smooth del Pezzo surface. Then if S is a cubic surface in P 3 with an Eckardt point, 3 4 if S is a cubic surface in P 3 without Eckardt points, 3 4 if K 2 S = 2 and | − K S | has a tacnodal curve, 5 6 if K 2 S = 2 and | − K S | has no tacnodal curves, 5 6 if K 2 S = 1 and | − K S | has a cuspidal curve, 1 if K 2 S = 1 and | − K S | has no cuspidal curves. 1 In [FO18], Fujita and Odaka introduced δ-invariant, which can be described as follows. For a sufficiently large and divisible integer k, consider a basis s 1 , · · · , s d k of the vector space H 0 (O X (−kK X )), where d k = h 0 (O X (−kK X )). For this basis, consider Q-divisor Any Q-divisor obtained in this way is called a k-basis type (anticanonical) divisor. Let δ k X = sup λ ∈ Q the log pair (X, λD) is log canonical for every k-basis type Q-divisor D ∼ Q −K X .
It turns out that δ(X) is extremely useful to test the K-stability of the Fano variety X.
In [PW18], Park and Won estimated δ-invariants of all smooth del Pezzo surfaces. In this paper, we give an alternative and more geometric approach to the same problem. Since [PW18] already gives a quite satisfying answer, we focus on cubic surfaces.
Our main result is the following Theorem 1.4. Let S be a smooth cubic surface in P 3 . Then δ(S) 6 5 . Corollary 1.5 ( [T90,PW18]). All smooth cubic surfaces in P 3 are uniformly K-stable.
For a smooth cubic surface S, it follows from [PW18,Theorem 4.9] that δ(S) 36 31 . The proof of Theorem 1.4 is completely different from the proof of [PW18,Theorem 4.9]. Moreover, our bound δ(S) 6 5 is slightly better . This paper is organized as follows. In Section 2, we present known results about divisors on smooth surfaces, and, as an illustration, we give a new proof of [PW18,Theorem 4.7]. In Section 3, we give various multiplicity estimates for basis type divisors on smooth cubic surfaces, which will be important to bound their δ-invariants in the proof of Theorem 1.4. These estimates also imply that δ-invariants of smooth cubic surfaces are at least 18 17 . In Section 4, we prove Theorem 1.4.
Acknowledgments. Kewei Zhang wants to thank his advisor Xiaohua Zhu for constant encouragement and support. The authors also thank Yanir Rubinstein for many helpful discussions. Ivan Cheltsov was partially supported by the Russian Academic Excellence Project "5-100". Kewei Zhang was supported by the China Scholarship Council No.201706010020. This paper was finished during the authors' visit to the Department of Mathematics at the University of Maryland, College Park. The authors appreciate its excellent environment and hospitality.

Basic tools
Let S be a smooth surface, let D be an effective R-divisor on it, let P be a point in S.
Let C be an irreducible curve on S. Write where a is a non-negative real number that is also denoted as ord C (D), and ∆ is an effective R-divisor on S whose support does not contain the curve C.
Lemma 2.2 ( [CRZ18,Proposition 3.3]). Suppose that a 1, the curve C is smooth at the point P , and mult P (∆) 1. If (S, D) is not log canonical at P , then Corollary 2.3. If a 1, the curve C is smooth at P , and the log pair (S, D) is not log canonical at P , then C · ∆ P > 1.
Let π : S → S be the blow up of the point P , and let E 1 be the exceptional curve of π. Denote by D the proper transform of D via π. Then

This implies
Corollary 2.4. The log pair (S, D) is log canonical at P if and only if the log pair S, D + mult P (D) − 1 E 1 is log canonical along the curve E 1 .
Thus, using Lemma 2.1 and Corollary 2.4, we obtain the following simple criterion.
If D is a Cartier divisor, then its volume is the number where the lim sup can be replaced by a limit (see [L04,Example 11.4.7]). Likewise, if D is a Q-divisor, we can define its volume using the identity vol(D) = vol λD λ 2 for an appropriate λ ∈ Q >0 . Then the volume vol(D) only depends on the numerical equivalence class of the divisor D. Moreover, the volume function can be extended by continuity to R-divisors. Furthermore, it is log-concave: for any pseudoeffective R-divisors D 1 and D 2 on the surface S. For more details about volumes of R-divisors, we refer the reader to [LM09,L04]. If D is not pseudoeffective, then vol(D) = 0. If the divisor D is nef, then This follows from the asymptotic Riemann-Roch theorem [L04]. If the divisor D is not nef, its volume can be computed using its Zariski decomposition [Z62, F79, P03, B09]. Namely, if D is pseudoeffective, then there exists a nef R-divisor N on the surface S such that where each C i is an irreducible curve on S with N · C i = 0, each a i is a non-negative real number, and the intersection form of the curves C 1 , . . . , C r is negative definite. Such decomposition is unique, and it follows from [BKS04,Corollary 3.2

This immediately gives
Corollary 2.7. Let Z 1 , . . . , Z s be irreducible curves on S such that D · Z i 0 for every i, and the intersection form of the curves Z 1 , . . . , Z s is negative definite. Then where b 1 , . . . , b s are (uniquely defined) non-negative real numbers such that for every j.
Corollary 2.8. Let Z be an irreducible curve on S such that Z 2 < 0 and D · Z 0. Then Let η : S → S be a birational morphism (possibly an identity) such that S is smooth. Fix a (non necessarily η-exceptional) irreducible curve F in the surface S. Let Theorem 2.9. Suppose that S is smooth del Pezzo surface, and D is a k-basis type divisor with k ≫ 1. Then where ǫ k is a small constant depending on k such that ǫ k → 0 as k → ∞.
Proof. This is a very special case of [FO18, Lemma 2.2].
This result plays a crucial role in the proof of Theorem 1.4. As a warm up, let us show how to use Theorem 2.9 to estimate δ-invariants of smooth del Pezzo surfaces of degree 1.
Theorem 2.10 ( [PW18,Theorem 4.7]). Let S be a smooth del Pezzo surface of degree 1. Then δ(S) 3 2 . Proof. Fix some rational number λ < 3 2 . Let D be a k-basis type divisor with k ≫ 1, and let P be a point in S. We have to show that the log pair (S, λD) is log canonical at P . By Lemma 2.1, it is enough to prove that Applying Theorem 2.9 with S = S, η = π and F = E 1 , we see that where ǫ k is a constant depending on k such that ǫ k → 0 as k → ∞. Let us compute τ (E 1 ). To do this, take a curve C ∈ | − K S | such that P ∈ C. Denote by C its proper transform on the surface S. If C is smooth at P , then C 2 = 0 and which implies that τ (E 1 ) = 1. In this case, we have Therefore, if C is smooth at P , then the log pair (S, λD) is log canonical at P for k ≫ 1.
To complete the proof, we may assume that C is singular at P . Then so that τ (E 1 ) = 2, since C 2 = −3. Using Corollary 2.8, we see that so that mult P (D) 5 6 + ǫ k . This gives δ(S) 6 5 . To get δ(S) 3 2 , we must work harder. Fix a point Q ∈ E 1 . By Corollary 2.5, to prove that (S, λD) is log canonical at P , it is enough to show that mult Q π * (D) = mult P D + mult Q D 2 λ .
Let σ : S → S be the blow up of the point Q. Denote by E 2 the exceptional curve of σ. Let η = π • σ. Applying Theorem 2.9 with F = E 1 , we see that Here, as above, the term ε k is a constant that depends on k such that ε k → 0 as k → ∞. Let C and E 1 be the proper transforms on S of the curves C and E 1 , respectively. Then the intersection form of the curves C and E 1 is negative definite. If Q ∈ C, then so that τ (E 2 ) = 3. In this case, using Corollary 2.8, we see that x 3, then Corollary 2.7 gives x 3, so that mult Q (π * (D)) 2 λ for k ≫ 1, because Likewise, if Q / ∈ C, then η * (−K S ) ∼ Q C + 2 E 1 + 2E 2 . so that τ (E 2 ) = 2. In this case, using Corollary 2.7, we deduce that so that mult Q (π * (D)) 2 λ for k ≫ 1. The following (simple) result can be very handy.
Lemma 2.11. In the assumptions and notations of Theorem 2.9, one has Proof. The assertion follows from the fact that vol(η Using (2.6), this result can be improved as follows: Lemma 2.12. In the assumptions and notations of Theorem 2.9, one has Proof. The required assertion follows from the proof of [F17, Proposition 2.1].
We will apply both Lemmas 2.11 and 2.12 to estimate the integral in Theorem 2.9 in the cases when it is not easy to compute.

Multiplicity estimates
Let S be a smooth cubic surface in P 3 , and let D be a k-basis type divisor with k ≫ 1. The goal of this section is to bound multiplicities of the divisor D using Theorem 2.9. As in Theorem 2.9, we denote by ǫ k a small number such that ǫ k → 0 as k → ∞.
Proof. Let us use assumptions and notations of Theorem 2.9 with η = Id S and F = L.
Let H be a general hyperplane section of the surface S that contains L. Then H = L + C, where C is an irreducible conic. Since C 2 = 0, we have τ (F ) = 1, so that Fix a point P ∈ S. Let π : S → S be the blowup of this point. Denote by E 1 the exceptional divisor of π. Fix a point Q ∈ E 1 . Let σ : S → S be the blowup of this point. Denote by E 2 the exceptional curve of σ. Let η = π • σ and F = E 2 . Let Applying Theorem 2.9, we get Let T P be the unique hyperplane section of the surface S that is singular at the point P . Then we have the following four possibilities: • T P = L 1 + L 2 + L 3 , where L 1 , L 2 and L 3 are lines such that P = L 1 ∩ L 2 ∩ L 3 ; • T P = L 1 + L 2 + L 3 , where L 1 , L 2 and L 3 are lines such that L 3 ∋ P = L 1 ∩ L 2 ; • T P = L + C, where L is a line and C is a conic such that P ∈ C ∩ L.
• T P is an irreducible cubic curve. We plan to bound the integral in (3.2) depending on the type of the curve T P and on the position of the point Q ∈ E 1 . First, we deal with the cases when Q is contained in the proper transform of the curve T P . We start with Lemma 3.3. Suppose that T P = L 1 + L 2 + L 3 , where L 1 , L 2 and L 3 are lines passing through P . Let L 1 , L 2 and L 3 be the proper transforms on S of the lines L 1 , L 2 and L 3 , respectively. Suppose that Q ∈ L 1 ∩ L 2 ∩ L 3 . Then Proof. We may assume that Q = L 1 ∩ E 1 . Denote by L 1 , L 2 , L 3 and E 1 the proper transforms on S of the curves L 1 , L 2 , L 3 and E 1 , respectively. Then the intersection form of the curves L 1 , L 2 , L 3 and E 1 is negative definite. Moreover, we have Thus, we conclude that τ (E 2 ) = 4. Now, using Corollary 2.7, we compute Then the required result follows from (3.2).
Lemma 3.4. Suppose that T P = L 1 + L 2 + L 3 , where L 1 , L 2 and L 3 are lines such that P = L 1 ∩ L 2 and P / ∈ L 3 . Let L 1 and L 2 be the proper transforms on S of the lines L 1 and L 2 , respectively. Suppose that Q = L 1 ∩ E 1 or L 2 ∩ E 1 . Then Proof. Denote by L 1 , L 2 , L 3 and E 1 the proper transforms on S of the curves L 1 , L 2 , L 3 and E 1 , respectively. Then Since the intersection form of the curves L 1 , L 2 , L 3 and E 1 is semi-negative definite, we conclude that τ (E 2 ) = 3. Then, using Corollary 2.7, we get Then the required result follows from (3.2).
Lemma 3.5. Suppose that T P = L + C, where L is a line, and C is an irreducible conic. Suppose that L and C meet transversally at P . Denote by L and C the proper transforms on S of the curves L and C, respectively. Suppose that Q = L ∩ E 1 . Then Proof. Denote by L, C and E 1 the proper transforms on S of the curves L, C and E 1 , respectively. Then Since the intersection form of the curves L, C and E 1 is negative definite, we conclude that τ (E 2 ) = 3. Moreover, using Corollary 2.7, we get x 3. Now the required assertion follows from (3.2).
Lemma 3.6. Suppose that T P = L + C, where L is a line, and C is an irreducible conic. Suppose that L and C meet transversally at P . Denote by L and C the proper transforms on S of the curves L and C, respectively. Suppose that Q = C ∩ E 1 . Then Proof. Denote by L, C and E 1 the proper transforms on S of the curves L, C and E 1 , respectively. Then Since the intersection form of the curves L, C and E 1 is negative definite, we conclude that τ (E 2 ) = 3. Moreover, using Corollary 2.7, we get Now the required assertion follows from (3.2).
Lemma 3.7. Suppose that T P = L + C, where L is a line and C is an irreducible conic. Suppose that L and C meet tangentially at P . Denote by L and C the proper transforms on S of the curves L and C, respectively. Suppose that Q = E 1 ∩ L ∩ C. Then Proof. Denote by L, C and E 1 the proper transforms on S of the curves L, L and E 1 , respectively. Then Since the intersection form of the curves L, C and E 1 is negative definite, we conclude that τ (E 2 ) = 4. Moreover, using Corollary 2.7, we get Then the required result follows from (3.2).
Lemma 3.8. Suppose that T P is an irreducible cubic. Let C be the proper transform of the curve C on the surface S. Suppose that Q ∈ C. Then Proof. Denote by C and E 1 the proper transforms on S of the curves C and E 1 , respectively. Then This gives τ (E 2 ) = 3, because the intersection form of the curves C and E 1 is negative definite. Using Corollary 2.7, we get Then the required result follows from (3.2). Now we consider the cases when Q is not contained in the proper transform of the singular curve T P on the surface S. We start with Lemma 3.9. Suppose that T P = L 1 + L 2 + L 3 , where L 1 , L 2 and L 3 are lines passing through P . Let L 1 , L 2 and L 3 be the proper transforms on S of the lines L 1 , L 2 and L 3 , Proof. Denote by L 1 , L 2 , L 3 and E 1 the proper transforms on S of the curves L 1 , L 2 , L 3 and E 1 , respectively. Then This gives τ (E 2 ) = 3, because the intersection form of the curves L 1 , L 2 , L 3 and E 1 is negative definite. Using Corollary 2.7, we get Then the required result follows from (3.2).
In the remaining cases, the pseudoeffective threshold τ (E 2 ) is not (always) easy to compute. There is a (birational) reason for this. To explain it, observe that the linear system | − K S | is free from base points and gives a morphism φ : S → P 2 . Taking its Stein factorization, we obtain a commutative diagram where α is a birational morphism, β is a double cover branched over a (possibly singular) quartic curve, and ρ is a linear projection from the point P . Here, the surface S is a (possibly singular) del Pezzo surface of degree 2. Note that the morphism α is biregular if and only if the curve T P is irreducible. Moreover, if T P is reducible, then α-exceptional curves are proper transforms of the lines on S that pass through P .
Let ι be the Galois involution of the double cover β. Then its action lifts to S. On the other hand, this action does not always descent to a (biregular) action of the surface S. Nevertheless, we can always consider ι as a birational involution of the surface S. This involution is known as Geiser involution. It is biregular if and only if P is an Eckardt point of the surface. In this case, the curve E 1 is ι-invariant. However, if P is not an Eckardt point, then ι(E 1 ) is the proper transform of the (unique) irreducible component of the curve T P that is not a line passing through P . In both cases, there exists a commutative diagram where S ′ is a smooth cubic surface in P 3 , which is isomorphic to the surface S via the involution τ , the morphism ν is the contraction of the curve ι(E 1 ), and ψ is a birational map given by the linear subsystem in | − 2K S | consisting of all curves having multiplicity at least 3 at the point P .
Let Q ′ = ν(Q) and P ′ = ν(ι(E 1 )). Denote by T ′ Q the unique hyperplane section of the cubic surface S ′ that is singular at Q ′ . If P is not an Eckardt point and Q is not contained in the proper transform of the curve T P , then Q ′ = P ′ . In this case, the number τ (E 2 ) can be computed using T ′ Q . This explains why the remaining cases are (slightly) more complicated.
Lemma 3.10. Suppose that T P = L 1 + L 2 + L 3 , where L 1 , L 2 and L 3 are lines such that P = L 1 ∩ L 2 and P / ∈ L 3 . Let L 1 , L 2 and L 3 be the proper transforms on S of the lines L 1 , L 2 and L 3 , respectively. Suppose that Q / ∈ L 1 ∪ L 2 . Then mult Q π * (D) 5 3 + ǫ k .
Proof. Denote by L 1 , L 2 , L 3 and E 1 the proper transforms on S of the curves L 1 , L 2 , L 3 and E 1 , respectively. Then which implies that τ (E 2 ) 2. Using Corollary 2.8, we see that provided that 0 x 2. However, we have τ (E 2 ) > 2, because the intersection form of the curves L 1 , L 2 , L 3 and E 1 is not semi-negative definite. This also follows from the fact that vol(η * (−K S ) − 2E 2 ) > 0. Recall that ν : S → S ′ is the contraction of the curve L 3 . We let L ′ 1 = ν( L 1 ), L ′ 2 = ν( L 2 ) and E ′ 1 = ν(E 1 ). Then L ′ 1 , L ′ 2 and E ′ 1 are coplanar lines on S ′ . Since Q ′ ∈ E ′ 1 , the line E ′ 1 is an irreducible component of the curve T ′ Q . Thus, either T ′ Q consists of three lines, or T ′ Q is a union of the line E ′ 1 and an irreducible conic.
Suppose that T ′ Q = E ′ 1 + Z ′ , where Z ′ is an irreducible conic on S ′ . Then Q ′ ∈ E ′ 1 ∩ Z ′ and Z ′ ∼ L ′ 1 + L ′ 2 , which implies that the conic Z ′ does not meet the lines L ′ 1 and L ′ 2 . Denote by Z the proper transform of the conic Z ′ on the surface S. We have This implies that τ (E 2 ) = 5 2 , because the intersection form of the curves Z, L 1 , L 2 and E 1 is semi-negative definite. Using this Q-rational equivalence and Corollary 2.7, we compute

Thus, a direct computation and (3.2) give
which gives the required assertion.
To complete the proof, we may assume that T ′ , which implies that the lines M ′ and N ′ do not meet the lines L ′ 1 and L ′ 2 . Denote by M and N the proper transforms on the surface S of the lines M ′ and N ′ , respectively.
Suppose that Q ′ is also contained in the line N ′ . This simply means that Q ′ is an Eckardt point of the surface S ′ . Then This gives τ (E 2 ) 3. In fact, we have τ (E 2 ) = 3 here, because the intersection form of the curves M , N , L 1 , L 2 , E 1 is negative definite. Using Corollary 2.7, we get

Now, direct computations and (3.2) give the required inequality.
To complete the proof the lemma, we have to consider the case Q ′ / ∈ N ′ . Then In particular, we see that τ (E 2 ) 5 2 . Using this Q-rational equivalence and Corollary 2.7, we compute As in the previous cases, we can find τ (E 2 ) and compute vol(η * (−K S ) −xE 2 ) for x > 5 2 . However, we can avoid doing this. Namely, note that the divisor E 1 + 2 N + M is nef and so that τ (E 2 ) 3. Therefore, using (3.2) and Lemma 2.11, we see that This finish the proof of the lemma.
Lemma 3.11. Suppose that T P = L + C, where L is a line and C is an irreducible conic. Denote by L and C the proper transforms on S of the curves L and C, respectively. Suppose that Q / ∈ L ∪ C. Then mult Q (π * (D)) 5 3 + ǫ k .
Proof. Denote by L, C and E 1 the proper transforms on S of the curves L, C and E 1 , respectively. Then so that τ (E 2 ) 2. Using Corollary 2.8, we see that Recall that ν : S → S ′ is the contraction of the curve C. Let L ′ = ν( L) and E ′ 1 = ν(E 1 ). Then L ′ is a line and E ′ 1 is a conic on S ′ such that P ′ ∈ L ′ ∩ E ′ 1 . First, we suppose that T ′ Q is irreducible. Denote by T Q the proper transform of the cubic T ′ Q on the surface S. Then T Q · E 1 = 0 and T Q · L = E 1 · L = 1.
Since L 2 = E 2 1 = −2 and T 2 Q = −1, we see that the intersection form of the curves L, T Q and E 1 is negative definite. On the other hand, we have This shows that τ (E 2 ) = 5 2 . Hence, using Corollary 2.7, we get , 2 x 17 7 , 4(5 − 2x) 2 , 17 7 x 5 2 . Then a direct calculation and (3.2) give where ℓ ′ is a line, and Z ′ is an irreducible conic. Denote by ℓ and Z the proper transforms on S of the curves ℓ ′ and Z ′ , respectively. We get which implies that τ (E 2 ) 5 2 . Using Corollary 2.7, we get , 2 x 5 2 . In particular, we have which implies that τ (E 2 ) > 5 2 . Observe that the divisor ℓ + 2 Z + L is nef and ℓ + 2 Z + L · η * (−K S ) − xE 2 = 9 − 3x, which implies that τ (E 2 ) 3. Thus, using (3.2) and Lemma 2.11, we get To complete the proof of the lemma, we may assume that T ′ 1 is a conic passing through Q ′ , we conclude that Q ′ is not contained in the line ℓ ′ . Note that ℓ ′ = L ′ , and the lines ℓ ′ , M ′ and N ′ do not pass through P ′ .
Denote by ℓ, M and N the proper transforms on S of the lines ℓ ′ , M ′ and N ′ , respectively. We get which implies that τ (E 2 ) 5 2 . In fact, we have τ (E 2 ) > 5 2 , because the intersection form of the curves ℓ, M , N, L and E 1 is not semi-negative definite. Nevertheless, we can use Corollary 2.7 to compute , 2 x 5 2 , so that, in particular, we have Observe that the divisor 2 ℓ + M + N is nef and which implies that τ (E 2 ) 3. Thus, using (3.2) and Lemma 2.12, we get The proof is complete.
Lemma 3.12. Suppose that T P is an irreducible cubic curve. Let C be its proper transform on the surface S. Suppose that Q / ∈ C. Then Proof. Denote by C and E 1 the proper transforms on S of the curves C and E 1 , respectively. Then Thus, using Corollary 2.8, we get vol(η * (−K S ) − xE 2 ) = 3 − x 2 2 provided that 0 x 2. Recall that ν : S → S ′ is the contraction of the curve C. Let E ′ = ν(E 1 ). Then E ′ 1 is an irreducible cubic curve that is singular at P ′ . Thus, the curve E ′ 1 is smooth at the point Q ′ , so that T ′ Q = E ′ 1 . One can easily check that T ′ Q does not contain P ′ . Suppose that T ′ Q is an irreducible cubic. Denote by T Q the proper transform of the curve T ′ Q on the surface S. We get E 2 1 = −2, T 2 Q = −1, E 1 · T Q = 1 and which implies that τ (E 2 ) = 5 2 . Using Corollary 2.7, we get x 5 2 . Then (3.2) and direct calculations give where ℓ ′ is a line and Z ′ is an irreducible conic. Denote by ℓ and Z the proper transforms on S of the curves ℓ ′ Q and Z ′ , respectively. We get Since the intersection form of the curves ℓ, Z and E 1 is semi-negative definite, we conclude that τ (E 2 ) = 5 2 . Using Corollary 2.7, we get Hence, using (3.2), we see that To complete the proof, we may assume that T ′ Q = ℓ ′ + M ′ + N ′ , where ℓ ′ , M ′ and N ′ are lines such that Q ′ ∈ M ′ ∩ N ′ . Denote by ℓ, M and N the proper transforms on S of the lines ℓ ′ , M ′ and N ′ , respectively. If Q ′ is contained in the line ℓ ′ , then and the intersection form of the curves ℓ, M, N and E 1 is negative definite, which implies that τ (E 2 ) = 3. In this case, Corollary 2.7 gives which implies the required inequality by (3.2).
To complete the proof, we may assume that Q ′ is not contained in ℓ ′ . Then the intersection form of the curves ℓ, M , N and E 1 is not semi-negative definite. Since we conclude that τ (E 2 ) > 5 2 . Moreover, using Corollary 2.7, we get , 2 x 5 2 .
In particular, we have Observe that the divisor 2 ℓ + M + N is nef and which implies that τ (E 2 ) 3. Thus, using (3.2) and Lemma 2.11, we get This completes the proof of the lemma.

Proof of the main result
In this section, we prove Theorem 1.4. Let S be a smooth cubic surface. We have to prove that δ(S) 6 5 . Fix a positive rational number λ < 6 5 . Let D be a k-basis type divisor. To prove Theorem 1.4, it is enough to show that, the log pair (S, λD) is log canonical for k ≫ 1. Suppose that this is not the case. Then there exists a point P ∈ S such that (S, λD) is not log canonical at P for k ≫ 1. Let us seek for a contradiction using results obtained in Section 3.
Let π : S → S be the blowup of the point P , and let E 1 be the exceptional divisor of the blow up π. Denote by D the proper transform of D via π. Then By Corollary 2.4, the log pair ( S, λ D + (λmult P (D) − 1)E 1 ) is not log canonical at some point Q ∈ E 1 . Thus, using Lemma 2.1, we see that Let σ : S → S be the blowup of the point Q, and let E 2 be the exceptional curve of σ. Denote by D and E 1 the proper transforms on S of the divisors D and E 1 , respectively. By Corollary 2.4, the log pair Let T P be the hyperplane section of the surface S that is singular at P . Then T P must be reducible. This follows from (4.1) and Lemmas 3.8 and 3.12.
Denote by T P the proper transform of the curve T P on the surface S. Then Q ∈ T P . This follows from (4.1) and Lemmas 3.10 and 3.11.
In the remaining part of this section, we will deal with the following four cases: (1) T P is a union of three lines passing through P ; (2) T P is a union of three lines and only two of them pass through P ; (3) T P is a union of line and a conic that intersect transversally at P ; (4) T P is a union of line and a conic that intersect tangentially at P . We will treat each of them in a separate subsection. We start with 4.1. Case 1. We have T P = L 1 + L 2 + L 3 , where L 1 , L 2 and L 3 are lines passing through the point P . We write λD = a 1 L 1 + a 2 L 2 + a 3 L 3 + Ω, where a 1 , a 2 and a 3 are nonnegative rational numbers, and Ω is an effective Q-divisor whose support does not contain L 1 , L 2 or L 3 . Then Denote by L 1 , L 2 and L 3 the proper transforms on S of the lines L 1 , L 2 and L 3 , respectively. We know that Q ∈ L 1 ∪ L 2 ∪ L 3 , so that we may assume that Q = L 1 ∩ E 1 .
Let Ω be the proper transform of the divisor Ω on the surface S, and let m = mult P (Ω). Then the log pair S, a 1 L 1 + Ω + a 1 + a 2 + a 3 + m − 1 E 1 is not log canonical at the point Q.
Denote by L 1 and Ω the proper transforms on S of the divisors L 1 and Ω, respectively. Then the log pair S, a 1 L 1 + Ω + a 1 + a 2 + a 3 + m − 1 E 1 + 2a 1 + a 2 + a 3 + m + m − 2 E 2 is not log canonical at the point O.
Denote by L 1 and L 2 the proper transforms on S of the lines L 1 and L 2 , respectively. We know that Q ∈ L 1 ∪ L 2 , so that we may assume that Q = L 1 ∩ E 1 . Let Ω be the proper transform of the divisor Ω on the surface S, and let m = mult P (Ω). Then the log pair S, a 1 L 1 + Ω + a 1 + a 2 + a 3 + m − 1 E 1 is not log canonical at the point Q.
Denote by L 1 and Ω the proper transforms on S of the divisors L 1 and Ω, respectively. Then the log pair S, a 1 L 1 + Ω + a 1 + a 2 + m − 1 E 1 + 2a 1 + a 2 + m + m − 2 E 2 is not log canonical at the point O. Then 2a 1 + a 2 + m + m − 2 < 1 by (4.10). Thus, using (4.11) and arguing as in Subsection 4.1, we see that O ∈ L 1 ∪ E 1 .
Denote by L and C the proper transforms on S of the curves L and C, respectively. We know that Q ∈ L ∪ C. Moreover, using (4.1) and Lemma 3.6, we see that Q = L ∩ E 1 .
Denote by Ω the proper transforms on S of the divisor Ω. Let m = mult P (Ω). Then the log pair S, a L + Ω + a + b + m − 1 E 1 is not log canonical at Q. Since a < 1, we can apply Corollary 2.3 to this log pair and the curve L. This gives L · Ω > 2 − a − b. Combining this with (4.13), we have λ + 2a − b > 2, so that (4.14) a > 2 + b − λ 2 2 − λ 2 .
Denote by L and Ω the proper transforms on S of the divisors L and Ω, respectively. Then the log pair S, a L + Ω + a + b + m − 1 E 1 + 2a + b + m + m − 2 E 2 is not log canonical at the point O. Note that 2a + b + m + m − 2 < 1 by (4.15). Thus, using (4.16) and arguing as in Subsection 4.1, we see that O ∈ L ∪ E 1 .

4.4.
Case 4. We have T P = L + C, where L is a line, and C is an irreducible conic that tangents L at the point P . We write λD = aL + bC + Ω, where a and b are nonnegative rational numbers, and Ω is an effective Q-divisor whose support does not contain L and C. Let m = mult P (Ω). Then (4.17) a + b + m > 1 by Lemma 2.1. Meanwhile, it follows from Lemma 3.1 that (4.18) a 1 5 9 + ε k λ < 1, where ε k is a small constant depending on k such that ε k → 0 as k → ∞. And also, we have (4.19) L · Ω = λ + a − 2b.
Denote by L and C the proper transforms on S of the curves L and C, respectively. We know that Q = L ∩ C. Denote by Ω the proper transforms on S of the divisor Ω. Then the log pair S, a L + b C + Ω + a + b + m − 1 E 1 is not log canonical at the point Q. Since a < 1 by (4.18), we may apply Corollary 2.3 to this log pair at Q with respect to the curve L. This gives L · Ω > 2 − a − 2b.
Denote by L, C and Ω the proper transforms on S of the divisors L, C and Ω, respectively. Then the log pair S, a L + b C + Ω + a + b + m − 1 E 1 + 2a + 2b + m + m − 2 E 2 is not log canonical at O. Moreover, it follows from (4.21) that 2a + 2b + m + m − 2 < 1. Thus, using (4.22) and arguing as in Subsection 4.1, we see that O ∈ L ∪ C ∪ E 1 .