Birational models of del Pezzo fibrations

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.

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…

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 $8$.

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。

Birational geometry of del Pezzo fibrations with terminal quotient singularities

Del Pezzo fibrations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo fibrations is a well studied question but smooth fibrations are not dense in moduli. Little is known about the rationality of the singular models. We prove birational rigidity, hence non-rationality, of del Pezzo fibrations with simple non-Gorenstein singularities satisfying the famous $K^2$-condition. We then apply this result to study embeddings of $\operatorname{PSL}_2(7)$ into the Cremona group.

On birational rigidity of singular del Pezzo fibrations of degree 1

We give a sufficient condition for birational superrigidity of del Pezzo fibrations of degree $1$ with only $\frac{1}{2} (1,1,1)$ singular points, generalizing the so called $K^2$-condition. As an application, we also prove that a del Pezzo fibrations of degree $1$ with only $\frac{1}{2} (1,1,1)$ singular points embedded in a toric $\mathbb{P} (1,1,2,3)$-bundle over $\mathbb{P}^1$ is birationally superrigid if and only if it satisfies the $K$-condition.

关于 del Pezzo 曲面的 $G$-双有理刚性

On $G$-birational rigidity of del Pezzo surfaces

本文证明，若代数闭域上的光滑 del Pezzo 曲面是 $H$-双有理刚性的，则它也是 $G$-双有理刚性的，从而在二维情形下正面回答了 Kollár 问题的几何版本。

六次del Pezzo曲面的双有理几何

Birational Geometry of sextic del Pezzo surfaces

该研究针对任意完美域上皮卡数为1的6次del Pezzo曲面，运用Galois上同调技术完成双正则分类与自同构群计算，解答了相关公开问题，证明其为唯一具无限柔韧性的立体曲面，并构造了双有理变换群的非平凡商。

Singular del Pezzo fibrations and birational rigidity

A known conjecture of Grinenko in birational geometry asserts that a Mori fibre space with the structure of del Pezzo fibration of low degree is birationally rigid if and only if its anticanonical class is an interior point in the cone of mobile divisors. The conjecture is proved to be true for smooth models (with a generality assumption for degree 3). It is speculated that the conjecture holds for, at least, Gorenstein models in degree 1 and 2. In this article, I present a (Gorenstein) counterexample in degree 2 to this conjecture.

Explicit biregular/birational geometry of affine threefolds: completions of A^3 into del Pezzo fibrations and Mori conic bundles

We study certain pencils of del Pezzo surfaces generated by a smooth del Pezzo surface S of degree less or equal to 3 anti-canonically embedded into a weighted projective space P and an appropriate multiple of a hyperplane H. Our main observation is that every minimal model program relative to the morphism lifting such pencil on a suitable resolution of its indeterminacies preserves the open subset P H â A^3. As an application, we obtain projective completions of A^3 into del Pezzo fibrations over P^1 of every degree less or equal to 4. We also obtain completions of A^3 into Mori conic bundles…

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。

Biregular and birational geometry of quartic double solids with 15 nodes

Three-dimensional del Pezzo varieties of degree 2 are double covers of projective space $\mathbb{P}^{3}$ branced in a quadric. In this paper we prove that if a del Pezzo variety of degree 2 has exactly 15 nodes then the corresponding quadric is a hyperplane section of the Igusa quartic or, equivalently, all such del Pezzo varieties are members of one particular linear system on the Coble fourfold. Their automorphism groups are induced from the automorphism group of Coble fourfold. Also we classify all $G$-birationally rigid varieties of such type.

Quotients of groups of birational transformations of cubic Del Pezzo fibrations

We prove that the group of birational transformations of a Del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, its subgroup generated by all connected algebraic subgroups is a proper normal subgroup.

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

We study the birational properties of geometrically rational surfaces from a derived categorical point of view. In particular, we give a criterion for the rationality of a del Pezzo surface over an arbitrary field, namely, that its derived category decomposes into zero-dimensional components. For del Pezzo surfaces of degree at least 5, we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of the surface from Brauer classes and Chern clas…

On stable rationality of Fano threefolds and del Pezzo fibrations

We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.

15-nodal quartic surfaces.I: quintic del Pezzo surfaces and congruences of lines in $\bbP^3$

We explain a classical construction of a del Pezzo surface of degree d = 4 or 5 as a smooth order two congruence of lines in 3-space whose focal surface is a quartic surface $X_{20-d}$ with 20-d ordinary double points. We also show that $X_{15}$ can be realized as a hyperplane section of the Castelnuovo-Richmond-Igusa quartic hypersurface. This leads to the proof of rationality of the moduli space of 15-nodal quartic surfaces. We discuss some other birational models of $X_{15}$: quartic symmetroids, 5-nodal quartic surfaces, 10-nodal sextic surfaces in $P^4$ and nonsingular surfaces of degree …

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.

On a rigidity criterion for del Pezzo fibrations over ${\mathbb P}^1$

We discuss the rigidity problem for Mori fibrations on del Pezzo surfaces of degree 1, 2 and 3 over ${\mathbb P}^1$ and formulate the following conjecture: such a del Pezzo fibration $V/{\mathbb P}^1$ is birationally rigid if and only if its quasi-effective and adjunction thresholds coincide. We prove the "only if" part of this conjecture.

del Pezzo对的模空间的双有理几何

Birational geometry of moduli space of del Pezzo pairs

本文研究d次光滑del Pezzo对模空间$P_d$的几何，从霍奇与GIT视角研究其紧化，得到Baily-Borel紧化类数，并针对d=8的一个分支构造Hassett-Keel-Looijenga模型，给出墙跨越的算术预测。

翻转与二次曲面在del Pezzo变种空间上的应用

Flips for spaces of quadrics on del Pezzo varieties

本文研究了del Pezzo变种空间上二次曲面的Hilbert方案的标准翻转构造，并推广了相关导出范畴分解的结果。

Quartic del Pezzo surfaces over function fields of curves

We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.

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.

The blow-up of $\mathbb{P}^4$ at 8 points and its Fano model, via vector bundles on a del Pezzo surface

Building on the work of Mukai, we explore the birational geometry of the moduli spaces M_{S,L} of semistable rank two torsion-free sheaves, with c_1=-K_S and c_2=2, on a polarized degree one del Pezzo surface (S,L); this is related to the birational geometry of the blow-up X of P^4 in 8 points. Our analysis is explicit and is obtained by looking at the variation of stability conditions. Then we provide a careful investigation of the blow-up X and of the moduli space Y=M_{S,-K_S}, which is a remarkable family of smooth Fano 4-folds. In particular we describe the relevant cones of divisors of Y,…

Classifying sections of del Pezzo fibrations, II

Let $X$ be a del Pezzo surface over the function field of a complex curve. We study the behavior of rational points on $X$ leading to bounds on the counting function in Geometric Manin's Conjecture. A key tool is the Movable Bend and Break Lemma which yields an inductive approach to classifying relatively free sections for a del Pezzo fibration over a curve. Using this lemma we prove Geometric Manin's Conjecture for certain split del Pezzo surfaces of degree $\geq 2$ admitting a birational morphism to $\mathbb P^2$ over the ground field.

Curves and line bundles on Del Pezzo surfaces

We extend the results of Pareschi on the constancy of the gonality and Clifford index of smooth curves in a complete linear system on Del Pezzo surfaces of degrees $\geq 2$ to the case of Del Pezzo surfaces of degree 1, where we explicitly classify the cases where the gonality and Clifford index are not constant. We also classify all cases of exceptional curves on Del Pezzo surfaces, which turn out to be the smooth plane curves and some other cases with Clifford dimension 3. Moreover, the property of being exceptional holds for all curves in the complete linear system. Furthermore, we relate t…

Gorenstein models of del Pezzo surfaces of degree 1 over Dedekind schemes

Let R be a Dedekind scheme, $ν$ its generic point, X and V del Pezzo surfaces of degree 1 over R that are Gorenstein Mori fiber spaces (as 3-folds germs over the ground field). We study birational maps $ϕ:X\dasharrow V$ over R which are isomorphisms over the generic point of R. We put down normal forms of such transformations (in suitable coordinates) and give some properties of X and V. In particular, we prove the uniqueness of a smooth model.

节点五次 del Pezzo 三维簇及其导出范畴

Nodal quintic del Pezzo threefolds and their derived categories

本文为节点五次 del Pezzo 三维簇构造川又型半正交分解，将其导出范畴分解为有限维代数的导出范畴，通过双有理映射到二次曲面纤维化实现。

del Pezzo 类型的标记 Artin--Schelter 曲面

Marked Artin--Schelter surfaces of del Pezzo types

本文引入 del Pezzo 类型的 Artin--Schelter 曲面（非交换曲面，含 del Pezzo 曲面为特例），并证明其标记模栈与亏格 1 曲线及若干线丛元组的模栈双有理。

Automorphisms of threefolds that can be represented as an intersection of two quadrics

In this paper we classify all potentially G-birationally rigid del Pezzo threefolds of degree 4 and their automorphism groups and prove the G-birational rigidity of one of them

四次del Pezzo表面的范畴Torelli定理

A categorical Torelli theorem for quartic del Pezzo surfaces

本文通过证明四次del Pezzo表面可由其Kuznetsov组件重建，解决了范畴Torelli问题，并验证了Kuznetsov组件的半正交不可分解性。

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.

On pliability of del Pezzo fibrations and Cox rings

We develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne-Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over $\mathbb{P}^1$, into other …

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.

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.

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…

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.