Computing invariants of singular del Pezzo surfaces
Abstract.
We prove new local inequality for divisors on surfaces and utilize it to compute invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type , , , , or are KählerEinstein.
We assume that all varieties are projective, normal, and defined over .
1. Introduction
Let be a Fano variety with at most quotient singularities (a Fano orbifold).
Theorem 1.1 ([37]).
If and is smooth, then
An important role in the proof of Theorem 1.1 is played by several holomorphic invariants, which are now known as invariants. Let us describe their algebraic counterparts.
Let be an effective divisor on the variety . Then the number
is called the log canonical threshold of the divisor (see [21, Definition 8.1]). Put
for every . For small , the number is usually not very hard to compute.
Example 1.2 ([28]).
If is a smooth surface in of degree , then
The number is denoted by in [38].
Remark 1.3.
If the variety is smooth, then it is proved by Demailly (see [6, Theorem A.3]) that
where is the invariant introduced by Tian in [36]. Put .
Conjecture 1.4 ([38, Question 1]).
There is an such that .
The proof of Theorem 1.1 uses (at least implicitly) the following result.
Note that there are many wellknown obstructions to the existence of Kähler–Einstein metrics on smooth Fano manifolds and Fano orbifolds (see [25], [14], [15], [34]).
Let us describe one more invariant that took its origin in [37].
Let be a linear system on the variety . Then the number
is called the log canonical threshold of the linear system (cf. [21, Theorem 4.8]). Put
for every . The number is denoted by in [8] and [41]. Note that
(1.7) 
and it follows from [21, Theorem 4.8] that for every .
Remark 1.8.
The importance of the number is due to the following conjecture.
Conjecture 1.9 (cf. [8, Theorem 2], [41, Theorem 1]).
Suppose that
for every . Then is Kähler–Einstein.
Example 1.10.
Suppose that is a smooth hypersurface in of degree . Then
for every by [2]. The equality holds the hypersurface contains a cone of dimension (see [2, Theorem 1.3], [2, Theorem 4.1], [13, Theorem 0.2]). Then
by Remark 1.8, [2, Remark 1.6], [2, Theorem 4.1], [2, Theorem 5.2] and [13, Theorem 0.2], because contains at most finitely many cones by [9, Theorem 4.2]. If is general, then
by [33], [3], [5]. Thus, if is general, then it is Kähler–Eisntein by Theorem 1.5.
The assertion of Conjecture 1.9 follows from [8, Theorem 2] and [41, Theorem 1] under an additional assumption that the KählerRicci flow on is tamed (see [8] and [41]).
Corollary 1.12.
Suppose that and
for every . Then is Kähler–Einstein.
Twodimensional Fano orbifolds are called del Pezzo surfaces.
Remark 1.13.
Del Pezzo surfaces with quotient singularities are not classified (cf. [20]). But
Del Pezzo surfaces with canonical singularities form a very natural class of del Pezzo surfaces.
Problem 1.14.
Describe all Kähler–Einstein del Pezzo surface with canonical singularities.
Recall that if is a del Pezzo surface with canonical singularities, then

either the inequality holds,

or one of the following possible cases occurs:

the equality holds and is a sextic surface in ,

the equality holds and is a quartic surface in ,

the equality holds and is a cubic surface in ,

the equality holds and is a complete intersection in of two quadrics.

Let us consider few examples to illustrate the expected answer to Problem 1.14.
Example 1.15.
Example 1.16.
Suppose that is a quartic surface in such that its singular locus consists of singular points of type or . Then is Kähler–Einstein by [16, Theorem 2].
Example 1.17.
Suppose that is a cubic surface in that is not a cone. Then
Example 1.18.
Suppose that is a complete intersection in of two quadrics. Then
Keeping in mind Examples 1.15, 1.16, 1.17 and 1.18, [4, Example 1.12] and [26, Table 1], it is very natural to expect that the following answer to Problem 1.14 is true (cf. Example 1.6).
Conjecture 1.19.
If the orbifold is a del Pezzo surface with at most canonical singularities, then the surface is Kähler–Enstein it satisfies one of the following conditions:

and consists of points of type , , , , , , or ,

and consists of points of type , or ,

and consists of points of type or ,

and consists of points of type ,

the surface is smooth and ,

either or .
In this paper, we prove the following result.
Theorem 1.20.
Suppose that is a sextic surface in . Then
for every if consists of points of type , , , , or .
Corollary 1.21.
Suppose that is a sextic surface in such that its singular locus consists of singular points of type , , , , or . Then is Kähler–Enstein.
It should be pointed out that Corollary 1.21 and Examples 1.15, 1.16, 1.17, 1.18 illustrate a general philosophy that the existence of Kähler–Enstein metrics on Fano orbifolds is related to an algebrogeometric notion of stability (see [11, Theorem 4.1], [39], [12]).
Remark 1.22.
If is a sextic surface in with canonical singularities, then either
or consists only of points of type and (see [40]).
What is known about invariants of del Pezzo surfaces with canonical singularities?
Theorem 1.23 ([3]).
If is a smooth del Pezzo surface, then .
Theorem 1.24 ([3], [31]).
If is a del Pezzo surface with canonical singularities, then
in the case when .
Theorem 1.25 ([31]).
If is a quartic surface in with canonical singularities, then
In this paper, we prove the following result (cf. Example 1.15).
Theorem 1.26.
Suppose that is a sextic surface in with canonical singularities, let be a natural double cover, and let be its branch curve in . Then
It should be pointed out that if is a del Pezzo surface with at most canonical singularities, then all possible values of the number are computed in [28], [29], [30].
Example 1.27.
If is a sextic surface in with canonical singularities, then

the surface has a singular point of type ,

the surface has a singular point of type ,

the surface has a singular point of type ,

the surface has a singular point of type , , , or ,

the following two conditions are satisfied:

the surface has no singular points of type , , , , , , or ,

there is a curve in that has a cusp at a point in of type ,


the following three conditions are satisfied:

the surface has no singular points of type , , , , , , or ,

there is no curve in that has a cusp at a point in of type ,

there is a curve in that has a cusp at a point in of type ,


the following three conditions are satisfied:

the surface has no singular points of type , , , , , , or ,

there is no curve in that have a cusp at a point in ,

there is a curve in that has a cusp,


there are no cuspidal curves in .
A crucial role in the proofs of both Theorems 1.26 and 1.20 is played by a new local inequality that we discovered. This inequality is a technical tool, but let us describe it now.
Let be a surface, let be an arbitrary effective divisor on the surface , let be a smooth point of the surface , let and be reduced irreducible curves on such that
and the divisor has a simple normal crossing singularity at the smooth point , let and be some nonnegative rational numbers. Suppose that the log pair
is not Kawamata log terminal at , but is Kawamata log terminal in a punctured neighborhood of the point .
Theorem 1.28.
Let be nonnegative rational numbers. Then
in the case when the following conditions are satisfied:

the inequality holds,

the inequalities hold,

the inequalities and hold,

either the inequality holds or
Corollary 1.29.
Suppose that
for some integer such that . Then
Proof.
To prove the required assertion, let us put
and let us check that all hypotheses of Theorem 1.28 are satisfied.
We have by assumption. We have
since . We have
since . We have and .
For the convenience of a reader, we organize the paper in the following way:

in Section 2, we collect auxiliary results,
2. Preliminaries
Let be a surface with canonical singularities, and let be an effective divisor on . Put
where is an irreducible curve, and . We assume that .
Suppose that is log canonical, but is not Kawamata log terminal.
Remark 2.1.
Let be an effective divisor on the surface such that
and the log pair is log canonical, where is a nonnegative rational number. Put
where is well defined and . Then . Suppose that . Put
and choose such that . Then and , but the log pair is not Kawamata log terminal.
Let be the locus of log canonical singularities of the log pair (see [6]).
Theorem 2.2 ([22, Theorem 17.4]).
If is nef and big, then is connected.
Take a point . Suppose that contains no curves that pass through .
Lemma 2.3.
Suppose that and . Then
Proof.
Let be a birational morphism, and is a proper transform of via . Then
where is an irreducible exceptional curve, and . We assume that .
Suppose, in addition, that the birational morphism induces an isomorphism
Remark 2.4.
The log pair is not Kawamata log terminal at a point in .
Suppose that is singular at , and either is a singular point of type for some , or the point is a singular point of type for some .
Lemma 2.5.
Suppose that . Then if
Proof.
Lemma 2.6.
Suppose that is a sextic surface in that has canonical singularities, and suppose that . Let be a positive rational number such that either
or and is not a curve in with a cusp at a point in of type . Then
the locus contains no points of type or , and .
3. Local inequality
The purpose of this section is to prove Theorem 1.28.
Let be a surface, let be an arbitrary effective divisor on the surface , let be a smooth point of the surface , let and be reduced irreducible curves on such that
and the divisor has a simple normal crossing singularity at the smooth point , let and be some nonnegative rational numbers. Suppose that the log pair
is not Kawamata log terminal at , but is Kawamata log terminal in a punctured neighborhood of the point . In particular, we must have and .
Let be nonnegative rational numbers such that

the inequality holds,

the inequalities hold,

the inequalities and holds,

either the inequality holds or
Lemma 3.1.
The inequalities and holds. The inequality
holds. The inequality holds. The inequalities
and hold.
Proof.
The inequality follows from the inequality . Then
because . Similarly, we see that , because
and . The inequality follows from the inequalities
because .
Let us show that the inequality
holds. Let be the line in given by the equation
and let be the line that is given by the equation
where are coordinates on . Then intersects the line at the point
and intersects the line at the point . But
which implies that if
where is the intersection point of the lines and . But
where . But
because , which implies that .
Finally, let us show that the inequality
holds. Let be the line in given by the equation
where are coordinates on . Then intersects the line at the point
and intersects the line at the point . But
which implies that if
where is the intersection point of the lines and . Note that
where .
To complete the proof, it is enough to show that the inequality
holds. This inequality is equivalent to the inequality
which is true, because and . ∎
Let us prove prove Theorem 1.28 by reductio ad absurdum. Suppose that the inequalities
hold. Let us show that this assumption leads to a contradiction.
Lemma 3.2.
The inequalities and hold.
Proof.
Put . Then is a positive rational number.
Remark 3.3.
The inequalities and hold.
Lemma 3.4.
The inequality holds.
Proof.
We know that and . Then