Log canonical thresholds of del Pezzo surfaces

Del Pezzo Zoo

We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose alpha-invariant of Tian is greater than 2/3.

Classification of Canonical Del Pezzo Orders

We classify del Pezzo non-commutative surfaces that are finite over their centres and have no worse than canonical singularities. Using the minimal model program, we introduce the minimal model of such surfaces. We first classify the minimal models and then give the classification of these surfaces in general. This presents a complementary result and method to the classification of del Pezzo orders over projective surfaces given by Chan and Kulkarni in 2003

A 1-dimensional component of K-moduli of del Pezzo surfaces

We explicitly construct a component of the K-moduli space of K-polystable del Pezzo surfaces which is a smooth rational curve.

Dynamic alpha-invariants of del Pezzo surfaces with boundary

The global log canonical threshold (or Tian's alpha-invariant) plays an important role in the geometry of Fano varieties. Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to determine when a smooth Fano variety X admits a Kahler-Einstein metric. It was conjectured that the existence of such a metric is equivalent to X being K-stable, an algebraic-geometric property. A crucial step in Donaldson's programme consists on finding a Kahler-Einstein metric with edge singularities of small angle along …

Computing $α$-invariants of singular del Pezzo surfaces

We prove new local inequality for divisors on surfaces and utilize it to compute $α$-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$, $\mathbb{A}_{2}$, $\mathbb{A}_{3}$, $\mathbb{A}_{4}$, $\mathbb{A}_{5}$ or $\mathbb{A}_{6}$ are Kähler-Einstein.

Exceptional del Pezzo hypersurfaces

We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$. As a corollary we obtain the existence of orbifold Kähler--Einstein metrics on many of them, and classify exceptional and weakly exceptional quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$.

Compact Moduli Spaces of Del Pezzo Surfaces and Kähler-Einstein metrics

We prove that the Gromov-Hausdorff compactification of the moduli space of Kahler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the existence of Kahler-Einstein metrics on smooth Del Pezzo surfaces and classifies the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.

K-不稳定 del Pezzo 曲面上的锥形 Kähler-Einstein 度量

Conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces

该研究确定了 K-不稳定 del Pezzo 曲面上锥形 Kähler-Einstein 度量锥角的最优上界，为理解此类度量的存在性提供关键约束。

Log canonical thresholds of Del Pezzo Surfaces in characteristic p

The global log canonical threshold of each non-singular complex del Pezzo surface was computed by Cheltsov. The proof used Kollár-Shokurov's connectedness principle and other results relying on vanishing theorems of Kodaira type, not known to be true in finite characteristic. We compute the global log canonical threshold of non-singular del Pezzo surfaces over an algebraically closed field. We give algebraic proofs of results previously known only in characteristic $0$. Instead of using of the connectedness principle we introduce a new technique based on a classification of curves of low degre…

反典范度 $\geq 5$ 的弱 del Pezzo 曲面的局部 delta 不变量

Local delta invariants of weak del Pezzo surfaces with the anti-canonical degree $\geq 5$

本文基于delta不变量对K-稳定性的判别准则，确定了反典范度 $\geq 5$ 的所有弱 del Pezzo 曲面的整个局部 delta 不变量。

Construction of good codes from weak Del Pezzo surfaces

We construct algebraic geometric codes from weak del Pezzo surfaces. The codes are associated to the anti-canonical class of the anti-canonical model and to the set of rational points of these models. Since we consider weak Del Pezzo surfaces, the anti canonical model is not smooth any more. This complicates the computation of the parameters of the codes; in particular we need to distinguish the Cartier divisors from the Weil ones.

Stable pair, tropical, and log canonical compact moduli of del Pezzo surfaces

We give a functorial normal crossing compactification of the moduli of smooth marked cubic surfaces entirely analogous to the Grothendieck-Knudsen compactification $M_{0,n} \subset \bar{M}_{0,n}$.

典范 del Pezzo 纤维化中的柱面

Cylinders in canonical del Pezzo fibrations

本文推广 Dubouloz 与 Kishimoto 的结果，利用一般纤维的次数及奇点类型，刻画具有典范奇点的 del Pezzo 纤维化包含垂直柱面的条件。

Moduli of double covers and degree one del Pezzo surfaces

Given a degree one del Pezzo surface with canonical singularities, the linear series generated by twice the anti-canonical divisor exhibits the surface as the double cover of the quadric cone branched along a sextic curve. It is natural to ask if this description extends to the boundary of a compactification of the moduli space of degree one del Pezzo surfaces. The goal of this paper is to show that this is indeed the case. In particular, we give an explicit classification of the boundary of the moduli space of anti-canonically polarized broken del Pezzo surfaces of degree one as double covers…

Classification of exceptional log Del Pezzo surfaces with delta=1

The exceptional log Del Pezzo surfaces with delta=1 are classified.

On higher-dimensional del Pezzo varieties

We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type A the dimension of non-conical del Pezzo varieties is bounded by $12 - d - r$, where $d$ is the degree and $r$ is the rank of the class group, and classify maximal del Pezzo varieties.

Existence of Kähler-Einstein metrics and multiplier ideal sheaves on del Pezzo surfaces

We apply Nadel's method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kähler-Einstein metric. In particular, all del Pezzo surfaces of degree $4,5$, or $6$ and certain special del Pezzo surfaces of lower degree are shown to have a Kähler-Einstein metric. This result is not new, but the proofs given in the present paper are less involved than earlier ones by Siu, Tian and Tian-Yau.

A note on Maximum Likelihood Estimation for cubic and quartic canonical toric del Pezzo Surfaces

This article focuses on the study of toric algebraic statistical models which correspond to toric Del Pezzo surfaces with Du Val singularities. A closed-form for the Maximum Likelihood Estimate of algebraic statistical models which correspond to cubic and quartic toric Del Pezzo surfaces with Du Val singular points is given. Also, we calculate the ML degrees of some toric Del Pezzo surfaces of degree less than or equal to six, which equals the degree of the surface in all the case but one, namely the quintic with two points of type $\mathbb{A}_1$.

Fibrations in sextic del Pezzo surfaces with mild singularities

We study sextic del Pezzo surface fibrations via root stacks.

Weak Approximation for General Degree Two del Pezzo Surfaces

We address weak approximation for certain del Pezzo surfaces defined over the function field of a curve. We study the rational connectivity of the smooth locus of degree two del Pezzo surfaces with two A1 singularities in order to prove weak approximation for degree two del Pezzo surfaces with square-free discriminant.

Weighted complete intersection del Pezzo surfaces

We classify codimension 2 well-formed and quasi-smooth weighted complete intersection del Pezzo surfaces.

Birational maps of del Pezzo fibrations

In this paper, we study rigidity of nonsingular del Pezzo fibrations over a germ of smooth curve.

关于具有du Val奇点的del Pezzo表面的核心正则性

On the coregularity of del Pezzo surfaces with du Val singularities

研究了具有du Val奇点的del Pezzo表面的核心正则性，发现正核心正则性表面对应于具有特殊性质的非平凡椭圆纤维化，且大多数表面核心正则性为0。

Unstable singular del Pezzo hypersurfaces with lower index

We give examples of K-unstable singular del Pezzo surfaces which are weighted hypersurfaces with index 2.

The maximal number of singular points on log del Pezzo surfaces

We prove that a del Pezzo surface with Picard number one has at most four singular points.

A note on del Pezzo fibrations of degree 1

We study del Pezzo fibrations of degree 1 with terminal singularities. A connection between singularities on del Pezzo surfaces of degree 1 and Kodaira's classification of elliptic singular fibers will be studied in this paper. By this investigation we obtain several results on birational maps of del Pezzo fibrations of degree 1 with terminal singularities.

Interpolation Problems: Del Pezzo Surfaces

We consider the problem of interpolating projective varieties through points and linear spaces. We show that del Pezzo surfaces satisfy weak interpolation.

On the moduli of degree 4 Del Pezzo surfaces

We study irreducibility of families of degree 4 Del Pezzo surface fibrations over curves.

Rational Points on Del Pezzo Surface of degree 1 and 2

We discuss the rational points on del Pezzo surface of degree 1 and 2 over any finite field $\mathbb F_q$, and give out the explicit equations of del Pezzo surfaces that have unique rational point.

K-stability of log del Pezzo hypersurfaces with index 2

We completely classify K-stability of log del Pezzo hypersurfaces with index 2.

Convergence of the Kähler-Ricci flow and multiplier ideal sheaves on del Pezzo surfaces

On certain del Pezzo surfaces with large automorphism groups, it is shown that the solution to the Kähler-Ricci flow with a certain initial value converges in $C^\infty$-norm exponentially fast to a Kähler-Einstein metric. The proof is based on the method of multiplier ideal sheaves.

Log del Pezzo surfaces with simple automorphism groups

In this paper we consider del Pezzo surfaces with only log terminal singularities admitting an action of a finite simple group.

Degenerations of Del Pezzo surfaces and the Gromov-Witten invariants of the Hilbert Scheme of conics

We investigate the characteristic numbers of Del Pezzo surfaces using degenerations.

Cylinders in del Pezzo surfaces with du Val singularities

We consider del Pezzo surfaces with du Val singularities. We'll prove that a del Pezzo surface $X$ with du Val singularities has a $-K_X$-polar cylinder if and only if there exist tiger such that the support of this tiger does not contain anti-canonical divisor. Also we classify all del Pezzo surfaces $X$ such that $X$ has not any cylinders.

On singularity types of del Pezzo surfaces with rational double points in positive characteristic

In this paper, we prove that a pair of the minimal resolution of a del Pezzo surface with rational double points whose general anti-canonical member is smooth and its exceptional divisor lifts to the Witt ring. We also classify a del Pezzo surface with rational double points whose anti-canonical members are all singular. As a corollary, we determine all singularity types of del Pezzo surfaces with rational double points which only appear in positive characteristic.

Combined count of real rational curves of canonical degree 2 on real del Pezzo surfaces with $K^2=1$

We propose two systems of "intrinsic" signs for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class $-2K$, but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.

Delta invariants of singular del Pezzo surfaces

We estimate $δ$-invariants of some singular del Pezzo surfaces with quotient singularities, which we studied ten years ago. As a result, we show that each of these surfaces admits an orbifold Kähler--Einstein metric.

具有四个log终端奇点的Del Pezzo表面

Del Pezzo surfaces with four log terminal singularities

本文研究了Picard数为一且具有四个log终端奇点的Del Pezzo表面的分类问题。

Del Pezzo surfaces with infinite automorphism groups

We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups.

Log del Pezzo surfaces with large volumes

We classify all of the log del Pezzo surfaces $S$ of index $a$ such that the volume $(-K_S^2)$ is larger than or equal to $2a$.

RDP del Pezzo surfaces with global vector fields in odd characteristic

We classify RDP del Pezzo surfaces with global vector fields over arbitrary algebraically closed fields of characteristic $p \neq 2$. In characteristic $0$, every RDP del Pezzo surface $X$ is equivariant, that is, ${\rm Aut}_X = {\rm Aut}_{\widetilde{X}}$, where $\widetilde{X}$ is the minimal resolution of $X$, hence the classification of RDP del Pezzo surfaces with global vector fields is equivalent to the classification of weak del Pezzo surfaces with global vector fields. In this article, we show that if $p \neq 2,3,5,7$, then it is still true that every RDP del Pezzo surface is equivariant…

Equivariant birational geometry of quintic del Pezzo surface

In this paper we prove that there are exactly two $G$-minimal surfaces which are $G$-birational to the quintic del Pezzo surface, where $G \cong C_5 \rtimes C_4$. These surfaces are the quintic del Pezzo surface itself and the surface $\mathbb{P}^1 \times \mathbb{P}^1$.

Klt del Pezzo surfaces which are not globally F-split

We construct a klt del Pezzo surface which is not globally F-split, over any algebraically closed field of positive characteristic.

The irreducibility of the spaces of rational curves on del Pezzo surfaces

We prove that the spaces of rational curves on del Pezzo surfaces are either irreducible or empty, with a unique exception.

Log del Pezzo Surfaces of Rank 2 and Cartier Index 3 with a Unique Singularity

Log del Pezzo surfaces play the role of the opposite of surfaces of general type. We will completely classify all the log del Pezzo surfaces of rank 2 and Cartier index 3 with a unique singularity.

Du Val del Pezzo曲面的自同构群

Automorphisms of Du Val del Pezzo surfaces

该研究针对自同构群为无限群的Du Val del Pezzo曲面，给出了其自同构群的描述结果。

界定几何整的 del Pezzo 曲面

Bounding geometrically integral del Pezzo surfaces

本文证明任意域上几何整正规 del Pezzo 曲面的有界性，给出不规则度锐界，并证明野生典范情形仅存在于特征 2，推广了 Tanaka 的结果并验证了 BAB 猜想。

$\mathbb{G}_a^3$-structures on del Pezzo fibrations

In this paper we prove that del Pezzo fibrations admit $\mathbb{G}_{a}^{3}$-structures if and only if they are $\mathbb{P}^2$-bundles over $\mathbb{P}^{1}$.

Almost del Pezzo manifolds

We classify almost del Pezzo manifolds in arbitrary dimension n, i.e., projective manifolds X with big and nef anticanonical bundle -K_X, such that -K_X is divisible by n-1.

Birational rigidity of $G$-del Pezzo threefolds of degree 2

In this paper we classify nodal rational non-$\mathbb{Q}$-factorial del Pezzo threefolds of degree 2 which can be $G$-birationally rigid for some subgroup $G\subset \operatorname{Aut}(X)$.

Potentially stably rational del Pezzo surfaces over nonclosed fields

We classify Galois actions on Picard lattices of del Pezzo surfaces of degrees 1,2, and 3 giving rise to minimal surfaces with no cohomological obstructions to stable rationality.

Log del Pezzo surfaces with not small fractional indices

For a log del Pezzo surface $S$, the fractional index $r(S)\in\mathbb{Q}_{>0}$ is the maximum of $r$ with which $-K_S$ can be written as $r$ times some Cartier divisor. We classify all the log del Pezzo surfaces $S$ with $r(S)\geq 1/2$, after the technique of Nakayama.

奇特征下del Pezzo曲面的自同构

Automorphisms of del Pezzo surfaces in odd characteristic

本文在奇正特征的代数闭域上，完成了del Pezzo曲面自同构群的分类工作。

低次数 del Pezzo 轨形分类 II

Classification of low degree del Pezzo orbifolds II

本文在 Campana 框架下，对次数小于 6 的 del Pezzo 轨形按边界分量进行分类，补充了先前不可约边界情形的工作。