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

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。

del Pezzo 曲面与圆锥丛的实纤维态射

Real-fibered morphisms of del Pezzo surfaces and conic bundles

本文基于 Ahlfors 关于实曲线的结果，利用拓扑环绕数刻画高维实代数簇的实纤维线性投影，并完整分类实 del Pezzo 曲面到射影平面的相关态射，同时探讨实圆锥丛。

Floor diagrams relative to a conic, and GW-W invariants of Del Pezzo surfaces

We enumerate, via floor diagrams, complex and real curves in the projective plane blown up in $n$ points on a conic. As an application, we deduce Gromov-Witten and Welschinger invariants of Del Pezzo surfaces. These results are mainly obtained using Li's degeneration formula and its real counterpart.

度数为1的极小德尔皮佐表面与有限域上的锥面的存在性

Existence of minimal del Pezzo surfaces of degree 1 with conic bundles over finite fields

本文研究有限域上度数为1的极小德尔皮佐表面的锥面结构，确定其存在性下界及特定类型不存在的q值，解决有限域上某些类型极小德尔皮佐表面的逆伽罗瓦问题。

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.

四次 del Pezzo 曲面上的圆锥曲线 10-网

On the 10-web by conics on the quartic del Pezzo surface

本文研究四次 del Pezzo 曲面上的圆锥曲线 10-网，证明 Bol 网及其阿贝尔恒等式的显著性质可推广至该网及其超对数关系。

del Pezzo曲面上的有理曲线空间通过二次曲面束

The spaces of rational curves on del Pezzo surfaces via conic bundles

本文利用Das-Lehmann-Tosteson等人发展出的同调筛方法，证明了分裂五次del Pezzo曲面在足够大的q下满足Peyre的全高度方法，同时建立了低次数分裂del Pezzo曲面上有理曲线计数函数的下界。

弱Del Pezzo曲面度数为4时切包的非负性

Positivity of tangent bundle of weak Del Pezzo surfaces of degree $4$

本文证明了度数至少为4的弱Del Pezzo曲面的切包几乎是 nef 的，通过利用由二次面束结构诱导的总双 VMRTs 进行证明。

Webs by conics on del Pezzo surfaces and hyperlogarithmic functional identities

For $d$ ranging from 2 to 6, we prove that the web by conics naturally defined on any smooth del Pezzo surface of degree $d$ carries an interesting functional identity whose components all are a certain antisymmetric hyperlogarithm of weight $7-d$. Our approach is uniform with respect to $d$ and at the end relies on classical results about the action of Weyl groups on the set of lines contained in the considered del Pezzo surface. This series of `del Pezzo's hyperlogarithmic functional identities' is a natural generalization of the famous and well-know 3-term and 5-term identities of the logar…

低次数del Pezzo曲面的弱逼近

Weak approximation for del Pezzo surfaces of low degree

本研究受Denef工作启发，采用算术满射性方法，证明满足一般性假设的带两个圆锥纤维化的曲面成立弱弱逼近，且次数为1或2的带圆锥纤维化的一般del Pezzo曲面也满足该性质。

A categorical invariant for geometrically rational surfaces with a conic bundle structure

We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on del Pezzo surfaces, this provides a categorical birational invariant for geometrically rational surfaces.

四度del Pezzo表面的理查德几何

Birational geometry of del Pezzo surfaces of degree 4

本文证明四度del Pezzo表面与自身同构，并回顾了A.N.Skorobogatov关于该类表面的biregular分类。

Classification of exceptional log Del Pezzo surfaces with delta=1

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

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.

Birational rigidity of orbifold degree 2 del Pezzo fibrations

Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ over the projective line with smooth total space satisfying the so-called $K^2$-condition are birationally rigid: their Mori fibre space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree $2$ are rather special, as for "most" families a general de…

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.

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.

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.

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.

Algorithms for singularities and real structures of weak Del Pezzo surfaces

In this paper we consider the classification of singularities (Du Val) and real structures (Wall) of weak Del Pezzo surfaces from an algorithmic point of view. It is well known that the singularities of weak Del Pezzo surfaces correspond to root subsystems. We present an algorithm which computes the classification of these root subsystems. We represent equivalence classes of root subsystems by unique labels. These labels allow us to construct examples of weak Del Pezzo surfaces with the corresponding singularity configuration. Equivalence classes of real structures of weak Del Pezzo surfaces a…

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.

Rational points of bounded height on general conic bundle surfaces

A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.

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。

Del Pezzo foliations with log canonical singularities

In this paper, we classify del Pezzo foliations on projective manifolds of rank at least 3 and with log canonical singularities in the sense of McQuillan.

$\mathbb{G}_a^3$-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.

Log canonical thresholds on Gorenstein canonical del Pezzo surfaces

We classify all the effective anticanonical divisors on weak del Pezzo surfaces. Through this classification we obtain the smallest number among the log canonical thresholds of effective anticanonical divisors on a given Gorenstein canonical del Pezzo surface.

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。

Birational models of del Pezzo fibrations

This is a preliminary version of the paper for the Lecture Notes Series.

具 du Val 奇点的 del Pezzo 曲面的复杂度

Complexity of del Pezzo surfaces with du Val singularities

该研究针对具 du Val 奇点的 del Pezzo 曲面，完成了其复杂度的计算工作。

无反典范极柱面的K-不稳定奇异del Pezzo曲面

K-unstable singular del Pezzo surfaces without anticanonical polar cylinder

该研究证明了存在既不具备K-半稳定性也不含反典范极柱面的奇异del Pezzo曲面，丰富了代数几何中相关曲面的性质研究。

Asymptotic Chow stability of toric Del Pezzo surfaces

In this short note, we study the asymptotic Chow polystability of toric Del Pezzo surfaces appear in the moduli space of Kähler-Einstein Fano varieties constructed in [OSS16].

Equations of log del Pezzo surfaces of index 2

Del Pezzo surfaces over C with log terminal singularities of index \le 2 were classified by Alekseev and Nikulin. In this paper, for each of these surfaces, we find an appropriate morphism to projective space. These morphisms enable us to describe del Pezzo surfaces of index \le 2 by equations in some weighted projective space. Obtained results are natural completions of similar results of Du Val, and Hidaka and Watanabe, describing del Pezzo surfaces of index 1.