On special quadratic birational transformations of a projective space into a hypersurface

We study transformations as in the title with emphasis on those having smooth connected base locus, called ??special??. In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by quadratic forms by showing that there are only four examples having general hyperplane sections of Severi varieties as base loci.     

Minitwistor spaces,Severi varieties,and Einstein–Weyl structure

In this article, 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 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 structure, then the real locus of the Severi variety becomes a positive definite Einstein–Weyl manifold. Moreover, we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves.     

Correction: On the Severi problem in arbitrary characteristic

In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi’s original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski’s theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.

     

Secondary fans and tropical Severi varieties

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 Severi variety cannot be a subfan of the corresponding secondary fan. The second theorem provides a partial converse, namely, we find conditions under which a cone of the secondary fan is fully contained in the tropical Severi variety. As a first application of these results, we also find a combinatorial formula for the tropical intersection multiplicities for secondary fans.     

Picard groups of severi varieties

We show that Severi varieties of sufficiently nodal plane curves have rational Picard group zero. The technique is to factor the natural map from the Severi variety to the moduli space of curves into a composition of fibrations each of which is well understood.     

The variety of reductions for a reductive symmetric pair

We define and study the variety of reductions for a complex reductive symmetric pair (G, θ), which is the natural compactification of the set of its Cartan subspaces. These varieties generalize the varieties of reductions for the Severi varieties studied by Iliev and Manivel, which are Fano varieties.     

Severi varieties

R. Hartshorne conjectured and F. Zak proved (cf [6,p.9]) that any smooth non-degenerate complex algebraic variety with satisfies denotes the secant variety of X; when X is smooth it is simply the union of all the secant and tangent lines to X). In this article, I deal with the limiting case of this theorem, namely the Severi varieties, defined by the conditions and . I want to give a different proof of a theorem of F. Zak classifying all Severi varieties. F. Zak proves that there exists only four Severi varieties and then realises a posteriori that all of them are homogeneous; here I will work in another direction: I prove a priori that any Severi variety is homogeneous and then deduce more quickly their classification, satisfying R. Lazarsfeld et A. Van de Ven's wish [6, p.18]. By the way, I give a very brief proof of the fact that the derivatives of the equation of Sec(X), which is a cubic hypersurface, determine a birational morphism of . I wish to thank Laurent Manivel for helping me in writing this article. Received in final form: 29 March 2001 / Published online: 1 February 2002     

Relative node polynomials for plane curves

We generalize the recent work of S.?Fomin and G.?Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and ?? nodes is given by a polynomial in d, provided ?? is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves that, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined ??relative node polynomial?? in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary ??, and use it to present explicit formulas for ????6. We also give a threshold for polynomiality, and compute the first few leading terms for any???.     

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?≤ g≤11, g?≠ 10.     

Singular Poisson-Kähler geometry of Scorza varieties and their secant varieties

Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kähler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.     

Severi inequality for varieties of maximal Albanese dimension

In this paper, we prove the general Severi inequality for varieties of maximal Albanese dimension. Suppose that \(X\) is an \(n\) -dimensional projective, normal, minimal and \(\mathbb {Q}\) -Gorenstein variety of general type in characteristic zero. If \(X\) is of maximal Albanese dimension, then \(K^n_X \ge 2 n! \chi (\omega _X)\) .     

The local–global principle for symmetric determinantal representations of smooth plane curves

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local–global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi–Brauer varieties and mod 2 Galois representations, and prove that the local–global principle holds for conics and cubics. We also construct counterexamples to the local–global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.     

Error-correcting codes on low rank surfaces

In this paper we construct some algebraic geometric error-correcting codes on surfaces whose Néron–Severi group has low rank. If the Néron–Severi group is generated by an effective divisor, the intersection of this surface with an irreducible surface of lower degree will be an irreducible curve, and this makes possible the construction of codes with good parameters. Such surfaces are not easy to find, but we are able to find surfaces with low rank, and those will give us good codes too.     

Volume functions of linear series

The volume of a Cartier divisor is an asymptotic invariant, which measures the rate of growth of sections of powers of the divisor. It extends to a continuous, homogeneous, and log-concave function on the whole Néron–Severi space, thus giving rise to a basic invariant of the underlying projective variety. Analogously, one can also define the volume function of a possibly non-complete multigraded linear series. In this paper we will address the question of characterizing the class of functions arising on the one hand as volume functions of multigraded linear series and on the other hand as volume functions of projective varieties. In the multigraded setting, inspired by the work of Lazarsfeld and Musta?? (Ann Inst Fourier (Grenoble) 56(6):1701–1734, 2006) on Okounkov bodies, we show that any continuous, homogeneous, and log-concave function appears as the volume function of a multigraded linear series. By contrast we show that there exists countably many functions which arise as the volume functions of projective varieties. We end the paper with an example, where the volume function of a projective variety is given by a transcendental formula, emphasizing the complicated nature of the volume in the classical case.     

Jordan Algebras Over Algebraic Varieties

General results on the module structure of Jordan algebras over locally ringed spaces are obtained. Albert algebras over a Brauer–Severi variety with associated central simple algebra of degree 3 are constructed using generalizations of the Tits process and the first Tits construction.     

Una formula di geometria numerativa

Riassunto Si riprende il concetto diintersezione semplice di ipersuperficie introdotto da Severi in legame a dei classici risultati di Kronecker, Vahlen. Ampia discussione, con esempi ed applicazioni, sulla base di una formula di equivalenza razionale qui dimostrata mediante l’anello di Chow.

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Summary The notion ofsimple intersection of huppersurfaces, introduced by Severi as a foundation to classical results of Kronecker, is resumed and discussed here on the basis of a formula of rational equivalence, which is proved using the Chow ring. Several examples and applications to the study of algebraic space curves, in particular to the quintic of Vahlen.

     

Universal polynomials for Severi degrees of toric surfaces

The Severi variety parameterizes plane curves of degree d$d$ with δ$\delta$ nodes. Its degree is called the Severi degree. For large enough d$d$, the Severi degrees coincide with the Gromov–Witten invariants of CP²$\mathbb{C}{\mathbb{P}}^2$. Fomin and Mikhalkin (2010) [10] proved the 1995 conjecture that for fixed δ$\delta$, Severi degrees are eventually polynomial in d$d$.     

Nikulin involutions on <Emphasis Type="Italic">K</Emphasis>3 surfaces

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the 11-dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on Morrison–Nikulin involutions, these are Nikulin involutions which interchange two copies of E ₈(−1) in the Néron Severi group. The second author is supported by DFG Research Grant SA 1380/1-1.