Almost del Pezzo manifolds

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.

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.

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。

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 $\mathbb{P}^{1}$.

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].

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…

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.

准线性作用于del Pezzo表面的理查德几何

Birational geometry of actions on del Pezzo surfaces

本文完成有限群在del Pezzo表面的正则一般自由作用的分类，解决了若干等变理查德几何中的开放问题。

奇异 del Pezzo 曲面上的刚性仿射锥

Rigid affine cones over singular del Pezzo surfaces

本文完全确定了指数为1的拟光滑 log del Pezzo 曲面中反典范极化柱的存在性。

Log canonical degenerations of del Pezzo surfaces in Q-Gorenstein families

We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings.

On exceptional collections of line bundles on weak del Pezzo surfaces

We study full exceptional collections of line bundles on surfaces. We prove that any full strong exceptional collection of line bundles on a weak del Pezzo surface of degree $\ge 3$ is an augmentation in the sense of L.Hille and M.Perling, while for some weak del Pezzo surfaces of degree $2$ the above is not true. We classify smooth projective surfaces possessing a cyclic strong exceptional collection of line bundles of maximal length: we prove that they are weak del Pezzo surfaces and find all types of weak del Pezzo surfaces admitting such a collection. We find simple criteria of exceptional…

Birational rigidity of del Pezzo fibrations with quotient singularities and its application

In this paper we give a criterion for birational rigidity of del Pezzo fibrations of degree 1 and 2 with only quotient singularities. As an application, we prove birational rigidity of suitable del Pezzo fibrations admitting an action of the Klein simple group.

Classification of log del Pezzo surfaces of index three

A normal projective non-Gorenstein log-terminal surface $S$ is called a log del Pezzo surface of index three if the three-times of the anti-canonical divisor $-3K_S$ is an ample Cartier divisor. We classify all of the log del Pezzo surfaces of index three. The technique for the classification based on the argument of Nakayama.

Classification of del Pezzo surfaces with $\frac{1}{3}(1,1)$- and $\frac{1}{4}(1,1)$-singularities

We classify all the del Pezzo surfaces with $\frac{1}{3}(1,1)$- and $\frac{1}{4}(1,1)$-singularities having no floating $(-1)$-curves into 39 types.

Stable rationality of del Pezzo fibrations of low degree over projective spaces

The main aim of this article is to show that a very general 3-dimensional del Pezzo fibration of degree 1,2,3 is not stably rational except for a del Pezzo fibration of degree 3 belonging to explicitly described 2 families. Higher dimensional generalizations are also discussed and we prove that a very general del Pezzo fibration of degree 1,2,3 defined over the projective space is not stably rational provided that the anticanonical divisor is not ample.

Movable Bend and Break for sections of del Pezzo fibrations

This is a report of the author's talk at Algebra Symposium 2019 held at Tohoku University. We discuss some improvements of Mori's Bend and Break for sections of del Pezzo fibrations over $\mathbb P^1$.

Brauer groups of singular del Pezzo surfaces

We describe the effect of rational singularities on the Brauer group of a surface, and compute the Brauer groups of all singular del Pezzo surfaces over an algebraically closed field.

Birational properties of pencils of del Pezzo surfaces of degree 1 and 2

We study the birational rigidity problem for smooth Mori fibrations on del Pezzo surfaces of degree 1 and 2. For degree 1 we obtain a complete description of rigid and non-rigid cases.

Hilbert functions of Cox rings of del Pezzo surfaces

To study syzygies of the Cox rings of del Pezzo surfaces, we calculate important syzygetic invariants such as the Hilbert functions, the Green-Lazarsfeld indices, the projective dimensions, and the Castelnuovo-Mumford regularities. Using these computations as well as the natural multigrading structures by the Picard groups of del Pezzo surfaces and Weyl group actions on Picard lattices, we determine the Betti diagrams of the Cox rings of del Pezzo surfaces of degree at most four.

On Gorenstein log del Pezzo Surfaces

In this paper, we first present the complete list of the singularity types of the Picard number one Gorenstein log del Pezzo surface and the number of the isomorphism classes with the given singularity type. Then we give out a method to find out all singularity types of Gorenstein log del Pezzo surface. As an application, we present the complete list of the Dynkin type of the Picard number two Gorenstein log del Pezzo surfaces. Finally we present the complete list of the singularity type of the relatively minimal Gorenstein log del Pezzo surface and the number of the isomorphism classes with t…

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…

del Pezzo曲面上的Ulrich三分法

Ulrich Trichotomy on del Pezzo surfaces

该研究通过射影簇上Ulrich丛与箭图表示的对应关系，证明了任意给定极化下部分del Pezzo曲面满足Ulrich三分法，为代数几何中相关曲面的丛理论研究提供了新结论。

Full Degree Two del Pezzo Surfaces over Small Finite Fields

Hirschfeld classified split del Pezzo surfaces of degree at least three whose points are all contained on the lines in the surface. We continue his work and begin the classification of split degree two del Pezzo surfaces over finite fields whose points are all on the fifty-six exceptional curves of the surfaces.