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(X), which is a cubic hypersurface, determine a birational morphism of P.
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.
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
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.
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.
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.
We generalize Zak's theorems on tangencies and on linear normality as well as Zak's definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, and we classify varieties for which the bound is attained.
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.
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.
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.
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.
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 $K^n_X \ge 2 n! χ(K_X)$
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.
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…
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.
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.
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…
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…
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 $\mathbb{P} ^N$.
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…
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…
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.
We study Brauer-Severi surface bundles over smooth projective varieties via root stacks, with a view towards applications to failure of stable rationality.
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…
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…
In this paper we define certain analogues of the volume of a divisor - called asymptotic cohomological functions - and investigate their behaviour on the Neron--Severi space. We establish that asymptotic cohomological functions are invariant with respect to the numerical equivalence of divisors, and that they give rise to continuous functions on the real Neron--Severi space. To illustrate the theory, we work out these invariants for abelian varieties, smooth surfaces, and certain homogeneous spaces.
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.
This article studies the relationship between tropical Severi varieties and secondary fans. In the case when tropical Severi varieties are hypersurfaces this relationship is very well known; specifically, in this case, a tropical Severi variety of codimension 1 is a subfan of the corresponding secondary fan. It was expected for some time that this continues to hold more generally, but Katz found a counterexample in codimension 2, showing that this relationship is more subtle. The two main results in this paper are as follows. The first theorem finds a simple condition under which a tropical Se…
We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities on an irregular variety $X$ are related to some natural defined birational invariants of a general fiber $F$ of the Albanese morphism of $X$. As applications, we provide a new lower bound of volumes of irregular threefolds, a sharp lower bound of varieties of maximal Albanese dimension and of general type, and show that the canonical model of such a variety with the minimal volume should be a flat double covers of a principally polarized ab…
In this paper we study the gonality of the normalizations of curves in the linear system $|H|$ of a general primitively polarized complex $K3$ surface $(S,H)$ of genus $p$. We prove two main results. First we give a necessary condition on $p, g, r, d$ for the existence of a curve in $ |H|$ with geometric genus $g$ whose normalization has a $g^ r_d$. Secondly we prove that for all numerical cases compatible with the above necessary condition, there is a family of \emph{nodal} curves in $|H|$ of genus $g$ carrying a $g^1_k$ and of dimension equal to the \emph{expected dimension} $\min\{2(k-1),g\…
