On Severi varieties and Moduli spaces of curves in arbitrary characteristic

The irreducibility of the generalized Severi varieties

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.

Severi, Scorza varieties and Jordan algebras

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.

No full exceptional collections on non-split Brauer--Severi varieties of dimension $\leq 3$

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 $\leq 3$.

Non-split Brauer-Severi varieties do not admit full exceptional collections

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.

椭圆曲线上的直纹面上的Severi簇

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。

On cubic torsors, biextensions and Severi-Brauer varieties over Abelian varieties

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}.

A characterization of secant varieties of Severi varieties among cubic hypersurfaces

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.

Severi 簇的几何

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。

通用布雷厄-塞弗里品种

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。

Severi varieties and their varieties of reductions

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.

On universal Severi varieties of low genus K3 surfaces

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.

Severi varieties on blow--ups of the symmetric square of an elliptic curve

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.

Severi's theorem for d-uple Veronese varieties

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.

Absolutely split locally free sheaves on proper $k$-schemes and Brauer--Severi varieties

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.

基变换型Enriques曲面及有理椭圆曲面上的Severi簇

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。

Severi inequality for varieties of maximal Albanese dimension

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 $K^n_X \ge 2 n! χ(K_X)$

Homotopy Rational Points of Brauer-Severi Varieties

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.

Motivic rigidity of Severi-Brauer varieties

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.

Severi varieties

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…

Incompressibility of quadratic Weil transfer of generalized Severi-Brauer varieties

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).

Severi varieties and self rational maps of K3 surfaces

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.

Lines on the secant cubic hypersurfaces of Severi varieties

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…

On the irreducibility of Severi varieties on K3 surfaces

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.

Ulrich bundles on Brauer--Severi varieties

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.

有限维塞弗里-布拉泽变元的自同构群的有限子群

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。

No phantoms in the derived category of curves over arbitrary fields, and derived characterizations of Brauer-Severi varieties

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…

Some results of regularity for Severi varieties of projective surfaces

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…

Decompositions of motives of generalized Severi-Brauer varieties

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.

On the Severi varieties of surfaces in P^3

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…

Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties

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 …

First Steps towards Hyper-desingularization through Brauer-Severi Varieties

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…

Intersection cohomology and Severi's varieties

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 $\mathbb{P} ^N$.

Generalized Clifford-Severi Inequality and the Volume of Irregular Varieties

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.

Severi-Brauer 簇的双有理等价

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。

Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties

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…

广义Severi-Brauer概型的半正交分解

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。

Severi-Brauer簇；一种几何处理

Severi-Brauer varieties; a geometric treatment

本文通过纯几何方法处理Severi-Brauer簇，不依赖中心单代数理论或Galois上同调，并提供了2026年的重大修订版。

Severi-Brauer簇自同构群的有限子群

Finite subgroups of automorphism groups of Severi--Brauer varieties

本文研究极小Severi-Brauer簇上有限子群的结构，并完整描述了维数为素数减一的这类簇上的所有有限子群。

Linear sections of the Severi variety and moduli of curves

We study the Severi variety $V_{d,g}$ of plane curves of degree $d$ and geometric genus $g$. Corresponding to every such variety, there is a one-parameter family of genus $g$ stable curves whose numerical invariants we compute. Building on the work of Caporaso and Harris, we derive a recursive formula for the degrees of the Hodge bundle on the families in question. For $d$ large enough, these families induce moving curves in $\bar{M}_g$. We use this to derive lower bounds for the slopes of effective divisors on $\bar{M}_g$. Another application of our results is to various enumerative problems …

Severi varieties and branch curves of abelian surfaces of type (1,3)

Let (A,L) be a principally polarized abelian surface of type (1,3). The linear system |L| defines a 6:1 covering of A onto P2, branched along a curve B of degree 18 in P2. The main result of the paper is that for general (A,L) the curve B irreducible, admits 72 cusps, 36 nodes or tacnodes, each tacnode counting as two nodes, 72 flexes and 36 bitangents. The main idea of the proof is to use the fact that for a general (A,L) of type (1,3) the closure of the Severi variety V in |L| is dual to the curve B in the sense of projective geometry. We investigate V and B via degeneration to a special abe…

Enriques 曲面上 Severi 簇的非空性

Nonemptiness of Severi varieties on Enriques surfaces

本文证明了在 $L$ 数值非 2-可除的一般极化 Enriques 曲面上，线性系统 $|L|$ 中所有不可约 $δ$-结点曲线的 Severi 簇均存在正则分支，从而解决了一个经典公开问题，并肯定地验证了 Pandharipande--Schmitt 的猜想。

On the torsion of Chow groups of Severi-Brauer varieties

In this paper, we generalize a result of Karpenko on the torsion in the second quotient of the gamma filtration for Severi-Brauer varieties to higher degrees. As an application, we provide a nontrivial torsion in higher Chow groups and the topological filtration of the associated generic variety and obtain new upper bounds for the annihilators of the torsion subgroups in the Chow groups of a large class of Severi-Brauer varieties. In particular, using the torsion in higher degrees, we show indecomposability of certain algebras.

Logarithmic abelian varieties, Part VII: Moduli

We construct the fine moduli space of log abelian varieties, which gives a compactification of the moduli space of abelian varieties.

K3曲面上Severi簇的不可约性

Irreducibility of Severi varieties on K3 surfaces

本文研究了一般本原极化K3曲面上Severi簇的连通性与不可约性，通过退化至Halphen曲面的方法，证明了当节点数$δ\leq g-1$时其闭包连通，并在$δ\leq g-4$时验证了不可约猜想。

Moduli of varieties of general type

This is a survey paper discussing the moduli problem for varieties of general type.

Moduli of logarithmic abelian varieties with PEL structure

We construct the fine moduli space of log abelian varieties with PEL structure, which gives a toroidal compactification of the moduli space of abelian varieties with PEL structure.

托里曲面上的Hurwitz空间和Severi簇的不可约性

The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces

本文通过热带几何方法，证明了在任意代数闭域上经典Hurwitz空间的不可约性，并建立了丰富托里曲面类（包括所有经典托里曲面）上Severi簇在任意特征下的不可约性。

关于环面曲面的Severi问题的一个注记

A note on the Severi problem for toric surfaces

本文推广了已有结果，通过商构造和风筝多边形给出了两类承认可约Severi簇的环面曲面，并引入不变量以提供分支数下界，同时建立了与单变量多项式拓扑分类的联系。

Birational isomorphisms between generalized Severi-Brauer varieties

The aim of this paper is to investigate the birational geometry of Generalized Severi-Brauer varieties. A conjecture of Amitsur states that two Severi-Brauer varieties $V(A)$ and $V(B)$ are birational if the underlying central simple algebras $A$ and $B$ are the same degree and generate the same cyclic subgroup of the Brauer group. We present a generalization of this conjecture to Generalized Severi-Brauer varieties, and show that in most cases we may reduce the new conjecture to the case where every subfield of the algebras is maximal, and in particular to the case where the algebras have pri…

Minitwistor spaces, Severi varieties, and Einstein-Weyl structure

In this paper we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space of smooth rational curves on a complex surface, given by N.Hitchin. As geometric objects naturally associated to Einstein-Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also we see that if the projective surface has an appropriate real structur…

Logarithmic abelian varieties, Part VI: Local moduli and GAGF

This is Part VI of our series of papers on log abelian varieties. In this part, we study local moduli and GAGF of log abelian varieties.

Models of Brauer-Severi surface bundles

We study Brauer-Severi surface bundles over smooth projective varieties via root stacks, with a view towards applications to failure of stable rationality.

塞弗里-布拉uer旗形 varieties 上的零次循环

Zero cycles on Severi--Brauer flag varieties

本文研究了塞弗里-布拉uer旗形 varieties 上的零次循环 Chow 群，证明了其在特定条件下为 (d, n/d)-torsion，并给出了新的实例和更严格的 torsion 绑定。

K-theory, LQEL manifolds and Severi varieties

We use topological K-theory to study non-singular varieties with quadratic entry locus. We thus obtain a new proof of Russo's Divisibility Property for locally quadratic entry locus manifolds. In particular we obtain a K-theoretic proof of Zak's theorem that the dimension of a Severi variety must be 2, 4, 8 or 16 and so resolve a conjecture of Atiyah and Berndt. We also show how the same methods applied to dual varieties recover the Landman parity theorem.