Alexei Elagin
In this paper we construct a full exceptional collection of sheaves on some family of log Del Pezzo surfaces, viewed as a smooth stack. These surfaces can be embedded as a quasi-smooth hypersurfaces into a certain weighted projective space, and they are not toric.
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The exceptional log Del Pezzo surfaces with delta=1 are classified.
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…
We investigate torsion exceptional sheaves on a weak del Pezzo surface of degree greater than two whose anticanonical model has at most $A_n$-singularities. We show that every torsion exceptional sheaf can be obtained from a line bundle on a $(-1)$-curve by acting spherical twist functors successively.
We show that simultaneous log resolutions of simply elliptic singularities can be constructed inside suitable stacks of principal bundles over elliptic curves. In particular, we give a direct geometrical construction of del Pezzo surfaces from the corresponding exceptional simple algebraic groups.
Let $X$ be a toric Del-Pezzo surface and let $Crit(W) \subset (\mathbb{C}^{\ast})^n$ be the solution scheme of the Landau-Ginzburg system of equations. Denote by $X^{\circ}$ the polar variety of $X$. Our aim in this work is to describe a map $L : Crit(W) \rightarrow Fuk_{trop}(X^{\circ})$ whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.
We prove that any numerically exceptional collection of maximal length, consisting of line bundles, on a smooth del Pezzo surface is a standard augmentation in the sense of L.Hille and M.Perling. We deduce that any such collection is exceptional and full.
Structure theorems for exceptional objects and exceptional collections of the bounded derived category of coherent sheaves on del Pezzo surfaces are established by Kuleshov and Orlov. In this paper we propose conjectures which generalize these results to weak del Pezzo surfaces. Unlike del Pezzo surfaces, an exceptional object on a weak del Pezzo surface is not necessarily a shift of a sheaf and is not determined by its class in the Grothendieck group. Our conjectures explain how these complications are taken care of by spherical twists, the categorification of $(-2)$-reflections acting on the…
Exceptional pairs on del Pezzo surfaces and spaces of compatible Feigin-Odesskii brackets
本文研究了del Pezzo曲面上的例外对与兼容Feigin-Odesskii括号空间的关系，证明了某些条件下的存在性和构造新例子。
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 $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$.
Given a del Pezzo surface of degree d between 1 and 6, possibly with rational double points, we construct a "tautological" holomorphic G-bundle over X, where G is a reductive group which is an appropriate conformal form of the simply connected complex linear group whose coroot lattice is isomorphic to the primitive cohomology of the minimal resolution of X. For example, in case d=3 and X is a smooth cubic surface, the rank 27 vector bundle over X associated to the G-bundle constructed above and the standard 27-dimensional representation of E_6 is a direct sum of the line bundles associated to …
Full exceptional collections for anticanonical log del Pezzo surfaces
本文受同调镜像对称启发，利用 McKay 对应与极小解消，为 Johnson–Kollár 对数 del Pezzo 曲面对应典范栈构造显式完全例外集，并刻画相关二次 del Pezzo 曲面。
Galois group of exceptional curves on the generic del Pezzo surface
研究了通用del Pezzo表面度数d的例外曲线的Galois作用，证明其为最大，并推导出在F_q(u)上立方面无Brauer-Manin障碍的结论。
Recently, de Thanhoffer de Volcsey and Van den Bergh showed that Grothendieck groups of "noncommutative Del Pezzo surfaces" with an exceptional sequence of length 4 are isomorphic to one of three types, the third one not coming from a commutative Del Pezzo surface. In this paper, we adapt the theory of noncommutative $\mathbb{P}^1$-bundles as appearing in the work of Van den Bergh and Nyman to produce a sheaf $\mathbb{Z}$-algebra whose associated Proj has an exceptional sequence of length 4 for which the Gram matrix is of this third type. We show that this noncommutative scheme is noetherian a…
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.
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.
We study sextic del Pezzo surface fibrations via root stacks.
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.
Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one
本文研究1次del Pezzo曲面上的例外曲线构型，证明了若一点包含在9条或更多例外曲线中则必为挠点，并给出7条相交时非挠点的反例及8条情形的部分结果。
We classify codimension 2 well-formed and quasi-smooth weighted complete intersection del Pezzo surfaces.
Generalized Eckardt points on del Pezzo surfaces of degree 1
本文研究度1 del Pezzo曲面上例外曲线的交点，分类了10条例外曲线相交的方式，构造了新曲面族，并探讨了寻找更多例子的策略。
In this paper, we study rigidity of nonsingular del Pezzo fibrations over a germ of smooth curve.
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。
We give examples of K-unstable singular del Pezzo surfaces which are weighted hypersurfaces with index 2.
We prove that a del Pezzo surface with Picard number one has at most four singular points.
We study complete exceptional collections of coherent sheaves over Del Pezzo surfaces, which consist of three blocks such that inside each block all Ext groups between the sheaves are zero. We show that the ranks of all sheaves in such a block are the same and the three ranks corresponding to a complete 3-block exceptional collection satisfy a Markov-type Diophantine equation that is quadratic in each variable. For each Del Pezzo surface, there is a finite number of these equations; the complete list is given. The 3-string braid group acts by mutations on the set of complete 3-block exceptiona…
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.
We consider the problem of interpolating projective varieties through points and linear spaces. We show that del Pezzo surfaces satisfy weak interpolation.
We study irreducibility of families of degree 4 Del Pezzo surface fibrations over curves.
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.
We completely classify K-stability of log del Pezzo hypersurfaces with index 2.
In this paper we consider del Pezzo surfaces with only log terminal singularities admitting an action of a finite simple group.
We investigate the characteristic numbers of Del Pezzo surfaces using degenerations.
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.
Let $X$ be a del Pezzo surface of degree one over an algebraically closed field $k$, and let $K_X$ be its canonical divisor. The morphism $φ$ induced by the linear system $|-2K_X|$ realizes $X$ as a double cover of a cone in $\mathbb{P}^3$ that is ramified over a smooth curve of degree 6. The surface $X$ contains 240 curves with negative self-intersection, called exceptional curves. We prove that for a point~$P$ on the ramification curve of $φ$, at most sixteen exceptional curves go through~$P$ in characteristic $2$, and at most ten in all other characteristics. Moreover, we prove that for a p…
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.
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。
We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups.
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 $2a$.
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…
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 $\mathbb{P}^1 \times \mathbb{P}^1$.
We construct a klt del Pezzo surface which is not globally F-split, over any algebraically closed field of positive characteristic.
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 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.
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。
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.
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 $\mathbb{P}^{1}$.
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.
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 $G\subset \operatorname{Aut}(X)$.
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.
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.
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.
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。
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。
This is a preliminary version of the paper for the Lecture Notes Series.
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。
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。
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].
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…
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.
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。
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