Delta Invariants of Singular del Pezzo Surfaces

We use the methods introduced by Cheltsov–Rubinstein–Zhang (Sel Math (N.S.) 25(2):25–34, 2019) to estimate δ -invariants of the seven singular del Pezzo surfaces with quotient singularities studied by Cheltsov–Park–Shramov (J Geom Anal 20(4):787–816, 2010) that have α -invariants less than 23 . As a result, we verify that each of these surfaces admits an orbifold Kähler–Einstein metric.


Introduction
Let S d be a quasismooth and well-formed hypersurface in P(a 0 , a 1 , a 2 , a 3 ) of degree d, where a 0 a 1 a 2 a 4 . Then S d is given by a quasihomogeneous polynomial equation of degree d: f x, y, z, t = 0 ⊂ P(a 0 , a 1 , a 2 , a 3 ) ∼ = Proj C x, y, z, t , where wt(x) = a 0 , wt(y) = a 1 , wt(z) = a 2 , and wt(t) = a 3 . Here, being quasismooth simply means that the above equation defines a hypersurface that is singular only at the origin in C 4 , which implies that S d has at most cyclic quotient singularities. On the other hand, being well-formed implies that see [15,Theorem 3.3.4], and [18, 6.14].
Put I = a 0 + a 1 + a 2 + a 3 − d and suppose that I is positive. Then S d is a del Pezzo surfaces with at most quotient singularities. If S d is smooth, then it always admits a Kähler-Einstein metric by Tian [33] (see also [7,25,30,35]). Singular del Pezzo surfaces with orbifold Kähler-Einstein metrics drew attention from Riemannian geometers because they may lift to Sasakian-Einstein 5-manifolds through S 1 -bundle structures. Through this passage, Boyer, Galicki and Nakamaye yielded a significant amount of examples towards classification of simply-connected Sasakian-Einstein 5-manifolds (see [5,6]).
In particular, if I = 1, then S d admits an orbifold Kähler-Einstein metric except possibly the case when (a 0 , a 1 , a 2 , a 3 , d) = (1, 3,5,7,15) and the defining equation of the surface S d does not contain yzt. Note that in the latter case one has by [8, Theorem 1.10] so that the criterion by the α-invariant could not be applied. Meanwhile, since 2010 we have witnessed dramatic developments in the study of the Yau-Tian-Donaldson conjecture concerning the existence of Kähler-Einstein metrics on Fano manifolds and stability. The challenge to the conjecture has been heightened by Chen, Donaldson, Sun and Tian who have completed the proof for the case of Fano manifolds with anticanonical polarizations [13,34]. Following this celebrated achievement, useful technologies have been introduced to determine whether given Fano varieties are Kähler-Einstein or not, via the theorem of Chen-Donaldson-Sun and Tian.
In this paper, we use δ-invariants to show that S d is Kähler-Einstein in these six cases as well: According to the similarity of the proofs, we handle the seven types of del Pezzo surfaces in Theorems 1.7 and 1.9 into three cases as follows: 3,5,7,15)  We will handle each of these cases separately in Sects. 3, 4 and 5, respectively; see Corollaries 3.3, 4.3 and 5.6. In Sect. 2, we will present some results that will be used in the proofs of Theorems 1.7 and 1.9.
Let us briefly explain how we estimate δ(S d ) in the proofs of Theorems 1.7 and 1.9. In our old paper [8], we developed a technique how to study possible singularities of log pairs (S d , D), where D is an effective Q-divisor on the surface S d such that  [8], to obtain the required estimates for δ(S d ). This approach was first used in [11] to estimate δ-invariants of the so-called asymptotically del Pezzo surfaces. Nevertheless, in our case we have an additional difficulty arising from the singularities of the surface S d , while all surfaces considered in [11] are smooth. It would be interesting to study the problem of existence of an orbifold Kähler-Einstein metric on S d in the remaining cases. In some of these cases, the del Pezzo surface S d is indeed not Kähler-Einstein. For instance, the surface S d does not admit an orbifold Kähler-Einstein metric in the case when I > 3a 0 . This follows from the obstruction found by Gauntlett et al. [17]. On the other hand, we expect the following to be true: We believe that this conjecture can be proved using a similar approach to the one we use in the proofs of Theorems 1.7 and 1.9.

Basic Tools
Let S be a surface with at most cyclic quotient singularities, let C be an irreducible reduced curve on S, let P be a point of the curve C, and let D be an effective R-divisor on the surface S. In this section, we present a few of well-known (local and global) results that will be used in the proof of Theorem 1.9. We start with To state an analogue of this result in the case when S is singular at P, recall that S has a cyclic quotient singularity of type 1 n (a, b) at the point P, where a and b are coprime positive integers that are also coprime to n. Thus, if n = 1, then P is a smooth point of the surface S. For n > 1, Corollary 2.2 can be generalized as follows: In general, the curve C may be contained in the support of the divisor D. Thus, we write where a is a non-negative real number, and is an effective R-divisor on S whose support does not contain the curve C. Then we have the following useful result:

Lemma 2.4 Suppose that a 1, the surface S is smooth at the point P, the curve C is also smooth at P, and the log pair (S, D) is not log canonical at P. Then
where C · P is the local intersection number of C and at P.
Proof This is a special case of a much more general result, known as the inversion of adjunction (see [28,31]).
The inversion of adjunction also holds for singular varieties. In our two-dimensional case, it can be stated as follows:

Lemma 2.5 Suppose that a 1, the log pair (S, C) is purely log terminal at P, and the log pair (S, D) is not log canonical at P. Then
Proof The required inequality follows from a more general version of the inverse of adjunction (see [28,31]). See also the proof of [8, Lemma 2.5].
By our assumption, the surface S has a cyclic quotient singularity of type 1 n (a, b) at the point P. Thus, locally near P, the surface S is a quotient of C 2 by the group Z n that acts on C 2 as where ω is a primitive nth root of unity. We can consider x and y as weighted coordinates around the point P.

Remark 2.6
The pair (S, C) has purely log terminal singularity at P if and only if C is given by x = 0 or y = 0 for an appropriate choice of weighted coordinates x and y. This follows from [28, Theorem 2.1.2], see also [20,Sect. 9.6]. Geometrically, this means that C is smooth at P, and its proper transform on the minimal resolution of singularities of the singular point P intersects the tail curve in the chain of exceptional curves. If (S, C) has purely log terminal singularities, then where we assume that S has a cyclic quotient singularity of index n O at the point O.
Let f : S → S be the weighted blow up of the point P with wt(x) = a and wt(y) = b, and let E be the exceptional curve of the morphism f . Then S has at most cyclic quotient singularities, one has E ∼ = P 1 , and the log pair ( S, E) has purely log terminal singularities. Moreover, the curve E has at most two singular points of the surface S. One of them is a singular point of type 1 a (n, −b), and another is a singular point of type 1 b (−a, n). Furthermore, we have If the curve C is locally given by x = 0 near the point P, then where C is the proper transform of the curve C on the surface S. For more properties of weighted blow ups and their defining equations, see [28,Sect. 3] or [2]. Denote by D the proper transform of the divisor D via f . Then for some non-negative rational number m. If C is not contained in the support of the divisor D, we can estimate m using where D · C and E · C can be computed in every case. Note that This implies

Proposition 2.7 The log pair (S, D) is log canonical at P if and only if the log pair
So far, we considered only local properties of the divisor D on the surface S. These properties will be used later to prove Theorem 1.9. However, the nature of this theorem is global so that we will need one global result that is due to Fujita and Odaka. To state it, we remind the reader of what the volume vol(D) of the R-divisor D is. If D is a Cartier divisor, then its volume is simply the number where the lim sup can be replaced by limit (see [21,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 positive rational number λ. One can show that the volume vol(D) only depends on the numerical equivalence class of the divisor D. Moreover, the volume function can be continuously extended to R-divisors (see [21] for details).
If D is not pseudoeffective, then vol(D) = 0. If D is pseudoeffective, its volume can be computed using its Zariski decomposition [3,29]. 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 a decomposition is unique, and it follows from [3, (2.8) Recall that D = aC + , where a is a non-negative real number, and is an effective divisor whose support does not contain the curve C. Let Then a τ . However, to prove Theorem 1.9, we have to find a better bound for a under an additional assumption that D is an ample Q-divisor of k-basis type for k 1 (for the definition, see [16, Definition 1.1] and the proof of Theorem 2.9 below). One such estimate is given by the following very special case of [16, Lemma 2.2].

Theorem 2.9
Suppose that D is a big Q-divisor of k-basis type for k 1. Then where k is a small constant depending on k such that k → 0 as k → ∞.
Proof Let us give a sketch of the proof that shows the nature of the required bound. First, recall from [16] that being k-basis type simply means that Here, we assume that k D is a Cartier divisor and k 0. Let M be a positive rational number such that M τ . We may assume that k M is an integer. Then there is a filtration of vector spaces Since the sections s 1 , . . . , s d k are linearly independent, we see that at most r 1 of them are contained in This immediately implies that the order of vanishing of the product s 1 · s 2 · s 3 · . . . · s d n at the curve C is at most Then we have a r 1 + r 2 + . . . + r k M−1 + r k M kr 0 .
As k → ∞, the right-hand side in this inequality converges to which gives the upper bound on a. For a detailed proof, we refer the reader to [16].

Corollary 2.10
Suppose that D is a big Q-divisor of k-basis type for k 1, and for some positive rational number μ. Then where k is a small constant depending on k such that k → 0 as k → ∞.
Proof Using Theorem 2.9, we get where k is a small constant depending on k such that k → 0 as k → ∞. But This implies the assertion.
By suitable coordinate changes, S 15 may be assumed to be given by  10  From now on, we suppose that b 1 = 0.

Proposition 3.2 Let D be an effective Q-divisor on S such that
Write D = aC x + , where a is a non-negative number, and is an effective Qdivisor on the surface S whose support does not contain the curve C x . Suppose also that a 8 21 . Then the log pair (S, 6 5 D) is log canonical.

Corollary 3.3
One has δ(S) 6 5 . Proof Let D be a Q-divisor of k-basis type divisor on S with k 0. Write D = aC x + , where a is a non-negative number, and is an effective Q-divisor on the surface S whose support does not contain the curve C x . By Corollary 2.10, we have a 8 21 for k 0. Thus, the log pair (S, 6 5 D) is log canonical for k 0 by Proposition 3.2. This implies that δ(S) 6 5 by Corollary 3.1.
To prove Proposition 3.2, we fix an effective Q-divisor D on the surface S such that Write D = aC x + , where a is a non-negative number, and is an effective Qdivisor on the surface S whose support does not contain the curve C x . Suppose also that a 8 21 . Let us show that the log pair (S, 6 5 D) is log canonical. Lemma 3. 4 The log pair (S, 6 5 D) is log canonical outside C x . Proof The required assertion follows from [8,Lemma 2.7]. For convenience of the reader, let us give the detailed proof here. Let P be a point in S \ C x . Since P / ∈ C x , there are complex numbers c 1 and c 2 such that P satisfies the following system of equations: Let P be the pencil of curves that is given by for [ν : μ] ∈ P 1 . Then the base locus of the pencil P consists of finitely many points. Moreover, by construction, the point P is one of them. Let C be a general curve in P. Then C · D 5 6 so that (S, 6 5 D) is log canonical at P by Corollary 2.2 if P is a smooth point of the surface S. This verifies the statement for S 15 .
For S 12 , we suppose that (S 12 , 6 5 D) is not log canonical at P. Then P must be one of the points O x , Q 1 ,Q 2 . Observe that the point P belongs to the curve C y cut by y = 0. Moreover, the curve C y is irreducible and the log pair (S 12 , 6 5 · 2 3 C y ) is log canonical. Thus, it follows from [9, Remark 2.22] that there exists an effective Q-divisor D on the surface S 12 such that the log pair (S 12 , 6 5 D ) is not log canonical at the point P, and the support of the divisor D does not contain the curve C y . However, which is impossible by Lemma 2.3 since (S 12 , 6 5 D ) is not log canonical at the point P. This completes the proof for S 12 .

Lemma 3.5
The log pair (S, 6 5 D) is log canonical at a point in C x \ {O t }. Proof Let P be a point in C x \ {O t }. Observe that P is a smooth point of the surface S, and C x is smooth at the point P. Note also that 6 5 a < 1. Thus, we can apply Lemma 2.4 to (S, 6 5 D) and the curve C x at the point P. Indeed, since the log pair (S, 6 5 D) must be log canonical at P.
Note that S 15 (resp. S 12 ) has singularity of type 1 7 (3, 5) (resp. 1 5 (3, 4)) at the point O t . In the chart defined by t = 1, the surface S 15 is given by and S 12 by Thus, in a neighborhood of the point O t , we may regard y and z as local weighted coordinates with wt(y) = 3 and wt(z) = 5 for S 15 and with wt(y) = 3 and wt(z) = 4 for S 12 .
Let f : S → S be the weighted blow up at the singular point O t with weights wt(y) = 3, wt(z) = 5 for S 15 and with weights wt(y) = 3, wt(z) = 4 for S 12 . Denote by E the exceptional curve of the blow up f . Then The surface S has two singular points in E. One is a point of type 1 3 (1, 1), and the other is a singular point of type and E ∼ = P 1 .
Let C x be the proper transform of the curve C x on the surface S. Then where c = 15 7 for S 15 and c = 12 5 for S 12 , and the intersection E ∩ C x consists of a single point, which is different from O 3 and O. Note that the curves E and C x intersect transversally at the point E ∩ C x .
Denote by be the proper transform of the Q-divisor on the surface S. Then for some non-negative rational number m. To estimate it, observe that Proof Suppose that the log pair (S, 6 5 D) is not log canonical at O t . Let us seek for a contradiction. Let λ = 6 5 . Then Thus, the log pair is not log canonical at some point Q ∈ E. Note that μ 1 because m We first apply Lemmas 2.4 or 2.5-(3.7) and the curve E at the point Q. Indeed, This shows that Q must be the intersection point of E and C x .
Applying Lemma 2.4 again, we see that However, these inequalities contradict our assumption a 8 21 . Therefore, the log pair (S, 6 5 Proposition 3.2 is completely verified.

Case B
The way to evaluate δ-invariants for Case B is almost same as that of Case A. In spite of this, we write the proof for the readers' convenience.
As in the previous section, we use S for the surfaces S 64 and S 82 if properties or conditions are satisfied by both the surfaces.
We may assume that the surface S 64 is given by the equation in P (7,15,19,32) and S 82 by the equation in P (7,19,25,41

Proposition 4.1 Let D be an effective Q-divisor on S such that
Write D = aC x + , where a is a non-negative number, and is an effective Qdivisor on the surface S whose support does not contain the curve C x . Suppose also that a 1 2 . Then the log pair (S, 19 18 D) is log canonical.
Proof Suppose also that a 1 2 . We first consider a point P that lies neither on C x nor on C y . Observe that P is a smooth point of the surface S. Since P / ∈ C x , there are complex numbers c 1 and c 2 such that P satisfies the following system of equations: Moreover, since P / ∈ C y , we have c 1 = 0. Let P be the pencil given by ν y 4 + c 1 x 5 t + μy y 3 + c 2 xz 2 = 0 on S 82 for [ν : μ] ∈ P 1 . The base locus of the pencil P consists of finitely many points. Furthermore, by construction, the point P is one of them. Let C be a general curve in P. Then It immediately follows from Corollary 2.2 that the log pair (S, 19 18 D) is log canonical outside C x and C y .
We next consider a point P on C x different from O z . Since a 1 2 , we apply Lemmas 2.4 and 2.5 to the log pair (S, 18 19 aC x + 18 19 ). Indeed, since on S 82 , the log pair (S, 19 18 D) must be log canonical at P. We now let P be a point on C y different from O z . Suppose that the log pair (S, 19 18 D) is not log canonical at the point P. Recall that (S 64 , 19 18 · 9 15 C y ) and (S 82 , 19 18 · 10 19 C y ) are log canonical, and the curve C y is irreducible. Thus, it follows from [9, Remark 2.22] that there exists an effective Q-divisor D on the surface S such that the log pair (S, 19 18 D ) is not log canonical at the point P and the support of the divisor D does not contain the curve C y . Observe This implies that the log pair (S, 19 18 D ) is log canonical at the point P. This contradicts our assumption. Thus, we see that (S, 19 18 D) is log canonical away from O z . Hence, to complete the proof of Proposition 4.1, we have to show that (S, 19 18 D) is log canonical at the point O z .
Recall that S 64 (resp. S 82 ) has singularity of type 1 19 (2, 3) (resp. 1 25 (2, 3)) at the point O z . In the chart z = 1, the surface S 64 is given by and S 82 by In a neighborhoods of the point O z , we can consider y and t as local weighted coordinates such that wt(y) = 2 and wt(t) = 3.
Let f : S → S be the weighted blow up at the singular point O z with weights wt(y) = 2 and wt(t) = 3. Denote by E the exceptional curve of the blow up f . Then The surface S has two singular points in E. One is a point of type 1 2 (1, 1) and the other is of type 1 3 (1, 1). Denote the former by O 2 and the latter by O 3 . Observe Let C x be the proper transform of the curve C x on the surface S. Then where c = 6 19 for S 64 and c = 6 25 for S 82 , and the intersection E ∩ C x consists of a single point different from O 2 and O 3 . Note that the curves E and C x intersect transversally.
Denote by be the proper transform of the Q-divisor on the surface S. Then for some non-negative rational number m. To estimate it, observe This implies m 19·25 for S 82 . We finally suppose that the log pair (S, 19 18 D) is not log canonical at O z . Let λ = 19 18 . Then Since This contradicts our assumption a 1 2 . The obtained contradiction completes the proof. 19 18 . Proof See the proof of Corollary 3.3.
As in the previous sections, we use S for all the surfaces S 45 , S 81 , and S 117 if properties or conditions are satisfied by all the surfaces.
The surface S is singular at the points Let C x be the curve in S that is cut out by x = 0. Then where L xz is the curve given by x = z = 0 and R x by x = z 2 − y 3 = 0 in the ambient weighted projective space. These two curves L xz and R x meets each other at the point O t . Also, we have , L xz · R x = 3 55 on S 117 .
Proof Suppose that D is of k-basis type with k 0. Theorem 2.9 implies that where k is a small constant depending on k such that k → 0 as k → ∞. Since and R 2 x < 0, we have vol(−K S − λL xz ) = 0 for λ 6 7 on S 45 , λ 8 7 on S 81 and λ 10 7 on S 117 . Similarly, using (5.1), we see that Then, using (5.1) again, we see that N · R x = 0 and N · L xz 0. Thus, we conclude that the divisor N is nef on the respective interval for λ. This shows that is the Zariski decomposition of the divisor −K S − λL xz . Hence, we have Finally, vol(−K S − λR x ) = 0 for λ > 6 7 on S 45 , for λ > 8 7 on S 81 , and for λ > 10 7 on S 117 since −K S −λR x is not pseudoeffective for these values λ. Thus, by Theorem 2.9, This yields the required bounds for b.
Now we prove the main assertion in this section.

Proposition 5.3 If a and b satisfy the bounds in Lemma 5.2 then the log pair (S, 65 64 D) is log canonical.
Proof We suppose that a and b satisfying the bounds in Lemma 5.2.
We first claim that the log pair (S, 65 64 D) is log canonical outside of C x and C y . This immediately follows from the same argument as in the beginning of the proof of Proposition 4.1 with the pencil P given by ν x 10 + c 1 y 7 + μy 4 z 2 + c 2 y 3 = 0 on S 45 , ν x 18 + c 1 y 7 + μy 4 z 2 + c 2 y 3 = 0 on S 81 , ν x 26 + c 1 y 7 + μy 4 z 2 + c 2 y 3 = 0 on S 117 ,