Ilya Tyomkin
In the current paper we prove that any Severi variety on a Hirzebruch surface contains a unique component parameterizing irreducible nodal curves of the given genus in characteristic zero.
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We prove that the locus of irreducible nodal curves on a given Hirzebruch surface F_k of given linear equivalency class and genus g is irreducible.
The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties) in his celebrated book "Topological Methods in Algebraic Geometry". The generalization for singular varieties is due to Brasselet-Schürmann-Yokura. Following the work of Weber, we investigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed points of the torus action and a formula for the local class in terms of the defining fan are mentioned. After this review part, we…
The classical Severi degree counts the number of algebraic curves of fixed genus and class passing through some general points in a surface. In this paper we study Severi degrees as well as several types of Gromov-Witten invariants of the Hirzebruch surfaces $F_k$, and the relationship between these numbers. To each Hirzebruch surface $F_k$ we associate an operator $\mathsf{M}_{F_k} \in \mathcal{H}[\mathbb{P}^1]$ acting on the Fock space $\mathcal{F}[\mathbb{P}^1]$. Generating functions for each of the curve-counting theories we study here on $F_k$ can be expressed in terms of the exponential …
For many classical moduli spaces of orthogonal type there are results about the Kodaira dimension. But nothing is known in the case of dimension greater than 19. In this paper we obtain the first results in this direction. In particular the modular variety defined by the orthogonal group of the even unimodular lattice of signature (2, 8m+2) is of general type if m is at least 5.
We consider the construction of Calabi-Yau varieties recently generalized to where the defining equations may have negative degrees over some projective space factors in the embedding space. Within such "generalized complete intersection" Calabi-Yau ("gCICY") three-folds, we find several sequences of distinct manifolds. These include both novel elliptic and K3-fibrations and involve Hirzebruch surfaces and their higher dimensional analogues. En route, we generalize the standard techniques of cohomology computation to these generalized complete intersection Calabi-Yau varieties.
Counting rational points on Hirzebruch-Kleinschmidt varieties over number fields
本文研究了数域上 Picard 秩为 2 的光滑射影分裂环面簇在 Arakelov 高度下的有理点渐近增长，证明其可分解为有限个不相交子簇并给出显式渐近公式，同时提供了包含 Hirzebruch 曲面的实例。
We study the tropicalizations of Severi varieties, which we call tropical Severi varieties. In this paper, we give a partial answer to the following question, ``describe the tropical Severi varieties explicitly.'' We obtain a description of tropical Severi varieties in terms of regular subdivisions of polygons. As an intermediate step, we construct explicit parameter spaces of curves. These parameter spaces are much simpler objects than the corresponding Severi variety and they are closely related to flat degenerations of the Severi variety, which in turn describes the tropical Severi variety.…
We study twisted forms of Schubert cells in generalized Severi-Brauer varieties, and show that the codimension $2$ Chow groups of these varieties are torsion free in certain cases, using the topological filtration on their K-theory
Counting rational points on Hirzebruch-Kleinschmidt varieties over global function fields
受Bourqui工作启发，本文研究了全局函数域上Picard秩为2的分裂环面簇的高度zeta函数，将其分解为子簇并给出精确解析性质，进而得到各子簇上大高度有理点计数的渐近公式。
We aim at giving a rigorous proof of the state-ments on the smoothness and the dimension of Severi varieties wherethere are gaps in the proofs in some standard literature. The method isa mixture of algebraic and analytic methods.
We classify morphisms from proper varieties to Brauer-Severi varieties, which generalizes the classical correspondence between morphisms to projective space and globally generated invertible sheaves. As an application, we study del Pezzo surfaces of large degree with a view towards Brauer-Severi varieties, and recover classical results on rational points, the Hasse principle, and weak approximation.
We classify products of symmetric powers of a Severi-Brauer variety, up-to stable birational equivalence. The description also includes Grassmannians and moduli spaces of genus 0 stable maps.
We give an inductive proof that the generalized Severi varieties -- the varieties which parametrize (irreducible) plane curves of given degree and genus, with a fixed tangency profile to a given line at several general fixed points and several mobile points -- are irreducible.
The Severi and Scorza varieties are the limiting cases of a theorem of Zak conjectured by Hartshorne. Zak also classified the Severi and Scorza varieties. Surprisingly enough, there are only 4 Severi varieties, one for each dimension 2,4,8 and 16 and they are homogeneous and very strongly linked with the 4 rank-3 Jordan algebras. In this article, I give several variations of Zak's classification theorem, proving a priori the homogeneity of Scorza varieties, and showing how it is possible to give a Jordan structure to the ambient space of a Scorza variety.
In this paper we study some new theories of characteristic homology classes for singular complex algebraic varieties. First we introduce a natural transformation T_{y}: K_{0}(var/X) -> H_{*}(X,Q)[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. T_{y} is a homology class version of the motivic measure corresponding to a suitable specialization of the well known Hodge poly…
In an earlier paper we showed that non-split Brauer--Severi curves do not admit full strong exceptional collections. In the present note we extend this observation and prove that there cannot exist full exceptional collections on non-split Brauer--Severi varieties of dimension $\leq 3$.
Recently, Novaković conjectured that non-split Brauer-Severi varieties do not admit full strong exceptional collections. In this short note, we explain how a stronger version of this conjecture follows easily from known results on noncommutative motives.
Severi Varieties on Ruled Surfaces over Elliptic Curves
研究一般椭圆曲线上的Atiyah直纹面上的Severi簇，证明其一般成员只有节点和通常三重点作为奇点。
We study the homogeneous irreducible Severi-Brauer varieties over an Abelian variety $A$. Such objects were classified by Brion, \cite{Bri}. Here we interpret that result within the context of cubical structures and biextensions for certain $\G_m$-torsors over finite subgroups of $A$. Our results can be seen as an instance of the theory developed by Breen, \cite{Breen:1983}, and Moret-Bailly, \cite{Moret-Bailly}.
It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.
The Geometry of Severi Varieties
本文总结了环面上 Severi 簇的几何性质，重点讨论了正特征下的可约与非节点分量，并利用热带几何工具将 Zariski 和 Harris 的定理推广至任意特征。
Universal Brauer-Severi varieties
本文提出了一种通用布雷厄-塞弗里品种的构造方法，并研究其几何性质及上同调群，同时重新诠释了判别回避结果。
We study the varieties of reductions associated to the four Severi varieties, the first example of which is the Fano threefold of index 2 and degree 5 studied by Mukai and others. We prove that they are smooth but very special linear sections of Grassmann varieties, and rational Fano manifolds of dimension $3a$ and index $a+1$, for $a=1, 2, 4, 8$. We study their maximal linear spaces and prove that through the general point pass exactly three of them, a result we relate to Cartan's triality principle. We also prove that they are compactifications of affine spaces.
We prove the irreducibility of universal Severi varieties parametrizing irreducible, reduced, nodal hyperplane sections of primitive K3 surfaces of genus g, with 3 \le g \le 11, g \neq 10.
This paper has been withdrawn by the author due to the gaps in the proofs of Proposition 2.2 and Proposition 3.2
The Hirzebruch $td_y(X)$ class of a complex manifold X is a formal combination of Chern characters of the sheaves of differential forms multiplied by the Todd class. The related $χ_y$-genus admits a generalization for singular complex algebraic varieties. The equivariant version of the Hirzebruch class can be developed as well. The general theory applied in the situation when a torus acts on a singular variety allows to apply powerful tools as the Localization Theorem of Atiyah and Segal for equivariant K-theory and Berline-Vergne formula for equivariant cohomology. We obtain a meaningful inva…
We prove that certain Severi varieties of nodal curves of positive genus on general blow-ups of the twofold symmetric product of a general elliptic curve are non-empty and smooth of the expected dimension. This result, besides its intrinsic value, is an important preliminary step for the proof of nonemptiness of Severi varieties on general Enriques surfaces in arXiv:2109.10735.
We characterize $d$-uple Veronese embeddings of finite-dimensional projective spaces. The easiest non-trivial instance of our theorem is the embedding of the projective plane in 5-dimensional projective space, a result obtained in 1901 by Severi when the underlying field is complex.
We classify absolutely split vector bundles on proper $k$-schemes. More precise, we prove that the closed points of the Picard scheme are in one-to-one correspondence with indecomposable absolutely split vector bundles. Furthermore, we apply the obtained results to study the geometry of (generalized) Brauer--Severi varieties.
Severi varieties on Enriques surfaces of base change type and on rational elliptic surfaces
本文证明了基变换型Enriques曲面上任意亏格曲线的Severi簇可数族非空，并构造了有理椭圆曲面上特殊超丰对数Severi簇的例子。
Let $X$ be a projective, normal, minimal and Gorenstein $n$-dimensional complex variety of general type. Suppose $X$ is of maximal Albanese dimension. We prove that $K^n_X \ge 2 n! χ(K_X)$
The paper is based on a talk. Complete exposition is given in "Equivariant Hirzebruch class for singular varieties". Starting from the classical theory we describe Hirzebruch class and the related Todd genus of a complex singular algebraic varieties. When the variety is equipped with an action of an algebraic torus we localize the Hirzebruch class at the fixed points of the action. We give some examples of computations.
We study homotopy rational points of Brauer-Severi varieties over fields of characteristic zero. We are particularly interested if a Brauer-Severi variety admitting a homotopy rational point splits. The analogue statement turns out to be true for open subvarieties of twisted hyperplane arrangements. In the complete case, the obstruction turns out to be the pullback of the Chern class of a generator of the Picard group along the homotopy rational point. Moreover, we will show that every Brauer-Severi variety over a $p$-adic field with $p$-primary period admits a homotopy rational point.
Let D be a central division algebra over a field F. We study in this note the rigidity of the motivic decompositions of the Severi-Brauer varieties of D, with respect to the ring of coefficients and to the base field. We first show that if the ring of coefficient is a field, these decompositions only depend on its characteristic. In a second part we show that if D remains division over a field extension E/F, the motivic decompositions of several Severi-Brauer varieties of D remain the same when extending the scalars to E.
R. Hartshorne conjectured and F. Zak proved that any n-dimensional smooth non-degenerate complex algebraic variety X in a m-dimensional projective space P satisfies Sec(X)=P if m<3n/2+2. In this article, I deal with the limiting case of this theorem, namely the Severi varieties, defined by the conditions m=3n/2+2 and Sec(X) different from P. I want to give a different proof of a theorem of F. Zak classifying all Severi varieties: I will prove that any Severi variety is homogeneous and then deduce their classification and the following geometric property : the derivatives of the equation of Sec…
Let X be the variety obtained by the Weil transfer with respect to a quadratic separable field extension of a generalized Severi-Brauer variety. We study (and, in some cases, determine) the canonical dimension, incompressibility, and motivic indecomposability of X. We determine the canonical 2-dimension of X (in the general case).
Self-rational maps of generic algebraic K3 surfaces are conjectured to be trivial. We relate this conjecture to a conjecture concerning the irreducibility of the universal Severi varieties parametrizing nodal curves of given genus and degree lying on some K3 surface. We also establish a number of numerical constraints satisfied by such non trivial rational maps, that is of topological degree >1.
The secant varieties of Severi varieties provide special examples of (singular) cubic hypersurfaces. An interesting question asks when a given cubic hypersurface is projectively equivalent to a secant cubic hypersurface. Inspired by the ''geometric'' Torelli theorem for smooth cubic hypersurfaces due to F. Charles, we study the geometry of lines on secant cubics and describe the Fano variety of lines. Then we verify the ''geometric'' Torelli theorem for the case of secant cubics. Namely, a cubic hypersurface is isomorphic to a secant cubic if and only if their Fano varieties of lines are isomo…
Hilbert schemes of suitable smooth, projective manifolds of low degree which are 3-fold scrolls over the Hirzebruch surface F_1 are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described.
Let $(S,L)$ be a polarized $K3$ surface of genus $p \geqslant 11$ such that $\mathrm{Pic}(S)=\mathbf{Z}[L]$, and $δ$ a non-negative integer. We prove that if $p\geqslant 4δ-3$, then the Severi variety of $δ$-nodal curves in $|L|$ is irreducible.
Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch $χ_y$-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.
We prove the existence of Ulrich bundles on any Brauer--Severi variety. In some cases, the minimal possible rank of the obtained Ulrich bundles equals the period of the Brauer--Severi variety. Moreover, we find a formula for the rank of an Ulrich bundle involving the period of the considered Brauer--Severi variety $X$, at least if $\mathrm{dim}(X)=p-1$ for an odd prime $p$. This formula implies that the rank of any Ulrich bundle on such a Brauer--Severi variety $X$ must be a multiple of the period.
We define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.
We prove generating series formulae for suitable twisted characteristic classes of symmetric products of a singular complex quasi-projective variety. More concretely, we study homology Hirzebruch classes for motivic coefficients, as well as for complexes of mixed Hodge modules. As a special case, we obtain a generating series formula for the (intersection) homology Hirzebruch classes of symmetric products. In some cases, the latter yields a similar formula for twisted homology L-classes generalizing results of Hirzebruch-Zagier and Moonen. Our methods also apply to the study of Todd classes of…
Finite subgroups of automorphism groups of Severi--Brauer varieties of prime degree
本文研究了素数维度的非平凡塞弗里-布拉泽变元的自同构群的有限子群分类，并构造了示例，证明了这些变元在特征不等于素数的域上不G-理性刚性。
In this paper we show that the derived category of Brauer-Severi curves satisfies the Jordan-Hölder property and cannot have quasi-phantoms, phantoms or universal phantoms. In this way we obtain that quasi-phantoms, phantoms or universal phantoms cannot exist in the derived category of smooth projective curves over a field $k$. Moreover, we show that a $n$-dimensional Brauer-Severi variety is completely characterized by the existence of a full weak exceptional collection consisting of pure vector bundles of length $n+1$, at least in characteristic zero. We conjecture that Brauer-Severi varieti…
Suppose an algebraic torus acts on a complex algebraic variety $X$. Then a great part of information about global invariants of $X$ are encoded in some data localized around the fixed points. The goal of this note is to present a connection between two approaches to localization for $C^*$-action. The homological results are related to $S^1$-action, while from $R^*_{>0}$-action we obtain a geometric decomposition. We study the resulting decompositions of Hirzebruch $χ_y$-genus and their relative versions. We show that via a limit process the second decomposition is obtained from the first one. …
For a linear system $|C|$ on a smooth projective surface $S$, whose general element is a smooth, irreducible curve, the Severi variety $V_{|C|, δ}$ is the locally closed subscheme of $|C|$ which parametrizes irreducible curves with only $δ$ nodes as singularities. In this paper we give numerical conditions on the class of divisors and upper-bounds on $δ$ ensuring that the corresponding Severi variety is everywhere smooth of codimension $δ$ in $|C|$ (regular, for short). In particular, we focus on surfaces of general type, since for such surfaces less is known than what is proven for other case…
Let p be a positive prime number and X be a Severi-Brauer variety of a central division algebra D of degree p^n, with n>0. We describe all shifts of the motive of X in the complete motivic decomposition of a variety Y, which splits over the function field of X and satisfies the nilpotence principle. In particular, we prove the motivic decomposability of generalized Severi-Brauer varieties X(p^m,D) of right ideals in D of reduced dimension p^m, m=0,1,...,n-1, except the cases p=2, m=1 and m=0 (for any prime p), where motivic indecomposability was proven by Nikita Karpenko.
The Severi variety V_{n,d} of a smooth projective surface S is defined as the subvariety of the linear system |O_S(n)|, which parametrizes curves with d nodes. We show that, for a general surface S of degree k in P^3 and for all n>k-1, d=0,...,dim(|O_S(n)|), there exists one component of V_{n,d} which is reduced, of the expected dimension dim(|O_S(n)|)-d. Components of the expected dimension are the easiest to handle, trying to settle an enumerative geometry for singular curves on surfaces. On the other hand, we also construct examples of reducible Severi varieties, on general surfaces of degr…
Let G be a semisimple affine algebraic group of inner type over a field F. We write C for the class of all finite direct products of projective G-homogeneous F-varieties. We determine the structure of the Chow motives with coefficients in a finite field of the varieties in C. More precisely, it is known that the motive of any variety in C decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions. In the case where G is the group of automorphisms of a given central simple F-algebra A, for any variety in the …
Given a Cayley-Hamilton smooth order A in a central simple algebra $Σ$, we determine the flat locus of the Brauer-Severi fibration of the smooth order. Moreover, we give a classification of all (reduced) central singularities where the flat locus differs from the Azumaya locus and show that the fibers over the flat, non-Azumaya points near these central singularities can be described as fibered products of graphs of projection maps, thus generalizing an old result of Artin on the fibers of the Brauer-Severi fibration over a ramified point. Finally, we show these fibers are also toric quiver va…
In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the complete case fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of …
Let $X^{2n}\subseteq \mathbb{P} ^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}π{_*}\mathbb{Q}$, where $π$ denotes the projection from the universal hyperplane family of $X^{2n}$ to ${(\mathbb{P} ^N)}^{\vee}$. We investigate the cohomology of the intersection cohomology complex $IC(R^{2n-1}π{_*}\mathbb{Q})$ over the points of a Severi's variety, parametrizing nodal hypersurfaces, whose nodes impose independent conditions on the very ample linear system giving the embedding in $\mathbb{P} ^N$.
Nested Hilbert Schemes on Hirzebruch surfaces and quiver varieties
本文证明Hirzebruch曲面$X_n$上的长度1嵌套Hilbert概型可表示为带关系的箭图簇，统一推广了$\mathbb C^2$上嵌套Hilbert概型与$X_n$上点Hilbert概型的已有结果。
We give a sharp lower bound for the selfintersection of a nef line bundle $L$ on an irregular variety $X$ in terms of its continuous global sections and the Albanese dimension of $X$, which we call the Generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibred irregular varieties. As a byproduct we obtain a lower bound for the volume of irregular varieties; when $X$ is of maximal Albanese dimension the bound is $Vol(X) \geq 2 n! χω_X$ and it is sharp.
It is well known that the category of quasi-coherent sheaves on a gerbe banded by a diagonalizable group decomposes according to the characters of the group. We establish the corresponding decomposition of the unbounded derived category of complexes of sheaves with quasi-coherent cohomology. This generalizes earlier work by Lieblich for gerbes over schemes whereas our gerbes may live over arbitrary algebraic stacks. By combining this decomposition with the semi-orthogonal decomposition for a projectivized vector bundle, we deduce a semi-orthogonal decomposition of the derived category of a fam…
Semiorthogonal decompositions for generalized Severi-Brauer schemes
本文利用保守下降方法为广义Severi-Brauer概型构造半正交分解，并推广到任意基概型上的旗簇，从而推广了Kapranov的经典结果。
Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schuermann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson L-classes, respectively). In this note we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms, and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We…
Birational equivalence of Severi-Brauer varieties
本文证明了指数不是素数幂的 Severi-Brauer 簇的 Amitsur 猜想。
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