Cascades of toric log del Pezzo surfaces of Picard number one

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.

Rank two aCM bundles on the del Pezzo threefold with Picard number 3

Let k be an algebraically closed field of characteristic 0. A del Pezzo threefold F with maximal Picard number is isomorphic to P^1xP^1xP^1, where P^1 is the projective line over k. In the present paper we completely classify locally free sheaves of rank 2 with vanishing intermediate cohomology over such an F. Such a classification extends similar results proved by E. Arrondo and L. Costa regarding del Pezzo threefolds with Picard number 1.

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.

On rank one log del Pezzo surfaces in characteristic different from two and three

We classify all log del Pezzo surfaces of Picard number one defined over algebraically closed fields of characteristic different from two and three. We also discuss some consequences of the classification. For example, we show that log del Pezzo surfaces of Picard number one defined over algebraically closed fields of characteristic higher than five admit a log resolution that lifts to characteristic zero over a smooth base.

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…

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。

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 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.

Anticanonical codes from del Pezzo surfaces with Picard rank one

We construct algebraic geometric codes from del Pezzo surfaces and focus on the ones having Picard rank one and the codes associated to the anticanonical class. We give explicit constructions of del Pezzo surfaces of degree 4, 5 and 6, compute the parameters of the associated anticanonical codes and study their isomorphisms arising from the automorphisms of the surface. We obtain codes with excellent parameters and some of them turn out to beat the best known codes listed on the database codetable.

无点的实 del Pezzo 曲面

Real del Pezzo surfaces without points

本文研究无点的实 del Pezzo 曲面，证明其在复数域上的 Picard 数至少为二。

德尔佩佐曲面之间的萨尔基索夫链的合成

Composition of Sarkisov links between del Pezzo surfaces

研究完美域上皮卡秩为一的双有理等价德尔佩佐曲面，核心方法是证明其双有理映射可分解为至多两个萨尔基索夫链的合成。

2次del Pezzo曲面丛与稳定有理性

Degree 2 del Pezzo surface bundles and stable rationality

本文研究函数域上2次del Pezzo曲面丛的算术性质，通过构造具有非平凡相对非分支上同调的例子，证明特殊化到该簇的簇不稳定有理性。

The spaces of rational curves on del Pezzo threefolds of degree one

We prove the irreducibility of moduli spaces of rational curves on a general del Pezzo threefold of Picard rank $1$ and degree $1$. As corollaries, we confirm Geometric Manin's conjecture and enumerativity of certain Gromov-Witten invariants for these threefolds.

Calabi--Yau 3-folds from projective joins of del Pezzo manifolds

In this paper, we will construct new examples of derived equivalent Calabi--Yau 3-folds with Picard number greater than one. We also study their mirror Calabi--Yau manifolds and find that they are given by Schoen's fiber products of suitable rational elliptic modular surfaces.

Gosset Polytopes in Picard groups of del Pezzo Surfaces

In this article, we research on the correspondences between the geometry of del Pezzo surfaces S_{r} and the geometry of Gosset polytopes (r-4)_{21}. We construct Gosset polytopes (r-4)_{21} in Pic S_{r}; Q whose vertices are lines, and we identify divisor classes in Pic S_{r} corresponding to (a-1)-simplexes, (r-1)-simplexes and (r-1)-crosspolytopes of the polytope (r-4)_{21}. Then we explain these classes correspond to skew a-lines, exceptional systems and rulings, respectively. As an application, we work on the monoidal transform for lines to study the local geometry of the polytope (r-4)_{…

Equivariant classification of Gorenstein open log del Pezzo surfaces with finite group actions

We classify equivariantly Gorenstein log del Pezzo surfaces with boundaries at infinity and with finite group actions such that the quotient surface modulo the finite group has Picard number one. We also determine the corresponding finite groups. Better figures are available upon request.

Bounding the volumes of singular weak log del Pezzo surfaces

We give an optimal upper bound for the anti-canonical volume of an $ε$-lc weak log del Pezzo surface. Moreover, we consider the relation between the bound of the volume and the Picard number of the minimal resolution of the surface. Furthermore we consider blowing up several points on a Hirzebruch surface in general position and give some examples of smooth weak log del Pezzo surfaces.

秩为一的Gorenstein del Pezzo表面的内射映射

Endomorphisms of rank one Gorenstein del Pezzo surfaces

本文研究了Picard秩为1的Gorenstein del Pezzo表面的内射映射性质，证明其与托尔品种的商结构有关，并对这类商进行了分类。

Standard models of degree $1$ del Pezzo fibrations

We construct a standard birational model (a model that has Gorenstein canonical singularities) for the three-dimensional del Pezzo fibrations ${ π\colon X \longrightarrow C }$ of degree $1$ and relative Picard number $1$. We also embed the standard model into the relative weighted projective space $\mathbb{P}_C (1,1,2,3)$. Our construction works in the $G$-equivariant category where $G$ is a finite group.

On the Picard number of divisors in Fano manifolds

Let X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in X. We consider the image H of N_1(D) in N_1(X) under the natural push-forward of 1-cycles. We show that the codimension c of H in N_1(X) is at most 8. Moreover if c>2, then either X=SxY where S is a Del Pezzo surface, or c=3 and X has a flat fibration in Del Pezzo surfaces onto a Fano manifold Y, such that the difference of the Picard numbers of X and Y is 4. We give applications to Fano 4-folds, to Fano varieties with pseudo-index >1, and to surjective morphisms whose source is Fano, having some high-dimensional…

Universal torsors of Del Pezzo surfaces and homogeneous spaces

Let Cox(S) be the homogeneous coordinate ring of the blow-up S of P^2 in r general points, i.e., a smooth Del Pezzo surface of degree 9-r. We prove that for r=6 and 7, Proj(Cox(S)) can be embedded into G/P, where G is an algebraic group with root system given by the primitive Picard lattice of S and P is a certain maximal parabolic subgroup in G.

Birational classification of pointless del Pezzo surfaces of degree 8

Let k be a perfect field. Recently J.-L. Colliot-Thélène showed that two pointless quadric surfaces over k are birationally equivalent if and only if they are isomorphic. We show that this result holds for arbitrary del Pezzo surfaces of degree $8$ with the Picard number $1$, and describe minimal surfaces birationally equivalent to a given pointless del Pezzo surface of degree $8$.

Projections of del Pezzo surfaces and Calabi--Yau threefolds

We study the syzygetic structure of projections of del Pezzo surfaces in order to construct singular Calabi-Yau threefolds. By smoothing those threefolds, we obtain new examples of Calabi-Yau threefolds with Picard group of rank 1. We also give an example of type II primitive contraction whose exceptional divisor is the blow-up of the projective plane at a point.

Smoothable del Pezzo surfaces with quotient singularities

We give a complete classification of del Pezzo surfaces with quotient singularities and Picard rank 1 which admit a Q-Gorenstein smoothing. There are 14 infinite families of toric examples. The surfaces in each family correspond to solutions of a Markov-type equation. The remaining surfaces are obtained as deformations of the toric surfaces or belong to a finite list of sporadic surfaces.

秩一 del Pezzo 曲面的分类。I. 高度 1 和 2。II. 具有椭圆边界的后代

Classification of del Pezzo surfaces of rank one. I. Height 1 and 2. II. Descendants with elliptic boundaries

本文作为系列研究首篇，以高度为不变量分类任意特征代数闭域上 Picard 秩一的 del Pezzo 曲面，完成高度至多 2 的分类并刻画具有椭圆边界后代的自然曲面类。

del Pezzo 流形上有理曲线空间的不可约性

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

本文通过研究赋值映射纤维，证明 Picard 秩 1、维数至少 4 的 del Pezzo 流形上有理曲线空间不可约，并由此推出几何 Manin 猜想。

K3 surfaces and log del Pezzo surfaces of index three

We use classification of non-symplectic automorphisms of K3 surfaces to obtain a partial classification of log del Pezzo surfaces of index three. We can classify those with "Multiple Smooth Divisor Property", whose definition we will give. Our methods include the definition of right resolutions of quotient singularities of index three and some analysis of automorphism-stable elliptic fibrations on K3 surfaces. In particular we find several log del Pezzo surfaces of Picard number one with non-toric singularities of index three.

具有环面作用的曲面的 Picard 指数

The Picard index of a surface with torus action

本文研究带环面作用的正规有理射影曲面，给出 Picard 群在除子类群中指数的计算公式，并据此分类 Picard 数为一且指数不超过 10,000 的 log del Pezzo 曲面。

一次单的单有理 del Pezzo 曲面

Unirational del Pezzo surfaces of degree one

本文构造了在 $\u211A$、$\u2115_5$、$\u2102(t)$ 上的算术皮卡秩为1的一次单有理del Pezzo曲面，并证明光滑实几何有理曲面在 $\u211D$ 上单有理当且仅当它有实点。

del Pezzo曲面上存在哪些有理二重点？

Which rational double points occur on del Pezzo surfaces?

本研究确定了任意特征代数闭域上任意次数与皮卡秩的RDP del Pezzo曲面的所有有理二重点构型，推广了Du Val的经典结果，并给出了含非紧有理二重点的1次RDP del Pezzo曲面的简化方程。

Quotients of del Pezzo surfaces

Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface and $G$ be a finite subgroup in $\operatorname{Aut}(X)$. In this paper we study when the quotient surface $X / G$ can be non-rational over $\Bbbk$. Obviously, if there are no smooth $\Bbbk$-points on $X / G$ then it is not $\Bbbk$-rational. Therefore under assumption that the set of smooth $\Bbbk$-points on $X / G$ is not empty we show that there are few possibilities for non-$\Bbbk$-rational quotients. The quotients of del Pezzo surfaces of degree $2$ and greater are considered in the author's previous papers. In this…

Del Pezzo surfaces with Du Val singularities

In this paper we study the problem of existence of orbifold Kaehler-Einstein metrics on del Pezzo surfaces of degree 1 with Du Val singular points. Moreover we compute global log canonical thresholds of del Pezzo surfaces of degree 1 with Du Val singularities and of del Pezzo surfaces of Picard rank 1 with Du Val singularities.

Line Bundles and Curves on a del Pezzo Order

Orders on surfaces provide a rich source of examples of noncommutative surfaces. Other than some existence results, very little is known about the various moduli spaces that can be associated to them. Even fewer examples have been explicitly computed. In this paper we compute the Picard and Hilbert schemes of an order on the projective plane ramified on a union of two conics. Our main result is that, upon carefully selecting the right Chern classes, the Hilbert scheme is a ruled surface over a genus two curve. Furthermore, this genus two curve is, in itself, the Picard scheme of the order.

The Cox ring of a Del Pezzo surface

Let $X_r$ be a smooth Del Pezzo surface obtained from $¶^2$ by blowing up $r \leq 8$ points in general position. It is well known that for $r \in \{3,4,5,6,7,8 \}$ the Picard group $\Pic(X_r)$ contains a canonical root system $R_r \in \{A_2 \times A_1, A_4, D_5, E_6, E_7, E_8 \}$. We prove some general properties of the Cox ring of $X_r$ ($r \geq 4$) and show its similarity to the homogeneous coordinate ring of the orbit of the highest weight vector in some irreducible representation of the algebraic group $G$ associated with the root system $R_r$.

Classification of Fano 4-folds with Lefschetz defect 3 and Picard number 5

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. If X contains a prime divisor D with rho(X)-rho(D)>2, then either X is a product of del Pezzo surfaces, or rho(X)=5 or 6. In this setting, we completely classify the case where rho(X)=5; there are 6 families, among which one is new. We also deduce the classification of Fano 4-folds with rho(X)>4 with an elementary divisorial contraction sending a divisor to a curve.

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.

五次del Pezzo三维流形上的秩二弱Fano丛

Rank two weak Fano bundles on del Pezzo threefolds of degree five

本文分类了五次del Pezzo三维流形上射影化为弱Fano的秩二向量丛，据此刻画其模空间的光滑性、不可约性与精细性，并推广到Picard秩为1的情形。

Del Pezzo surfaces with a single 1/k(1,1) singularity

Inspired by the recent progress by Coates-Corti-Kasprzyk et al. on Mirror Symmetry for del Pezzo surfaces, we show that for any positive integer k the deformation families of del Pezzo surfaces with a single 1/k(1,1) singularity (and no other singular points) fit into a single cascade. Additionally we construct models and toric degenerations of these surfaces embedded in toric varieties in codimension less than or equal two. Several of these directly generalise constructions of Reid-Suzuki (in the case k=3). We identify a root system in the Picard lattice, and in light of the work of Gross-Hac…

A boundedness result for toric log Del Pezzo surfaces

In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index $\ell$. This upper bound turns out to be a quadratic polynomial in the variable $\ell$.

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。

On Fano foliations 2

In this paper we study mildly singular del Pezzo foliations on complex projective manifolds with Picard number one

Quotients of del Pezzo surfaces of degree 2

Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface of degree~$2$ and $G$ be a group acting on $X$. In this paper we study $\Bbbk$-rationality questions for the quotient surface $X / G$. If there are no smooth $\Bbbk$-points on $X / G$ then $X / G$ is obviously non-$\Bbbk$-rational. Assume that the set of smooth $\Bbbk$-points on the quotient is not empty. We find a list of groups, such that the quotient surface can be non-$\Bbbk$-rational. For these groups we construct examples of both $\Bbbk$-rational and non-$\Bbbk$-rational quotients of both $\Bbbk$-rational and non…

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.

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.