相关论文: Geometry of theta divisors --- a survey
We study the birational geometry of some moduli spaces of abelian varieties with extra structure: in particular, with a symmetric theta structure and an odd theta characteristic. For a $(d_1,d_2)$-polarized abelian surface, we show how the…
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the…
The singularities of theta divisors have played an important role in the study of algebraic varieties. This paper surveys some of the recent progress in this subject, using as motivation some well known results, especially those for…
To any closed subvariety $Y$ of a complex abelian variety one can attach a reductive algebraic group $G$ which is determined by the decomposition of the convolution powers of $Y$ via a certain Tannakian formalism. For a theta divisor $Y$ on…
The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of…
The Andreotti-Mayer locus is a subset of the moduli space of principally polarized abelian varieties, defined by a condition on the dimension of the singular locus of the theta divisor. It is known that the Jacobian locus in the moduli…
The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the…
This is an expository paper about the topics listed in the title.
Inspired by a theorem of Gruson-Lazarsfeld-Peskine bounding the Castelnuovo-Mumford regularity of curves in projective spaces, we bound the Theta-regularity of curves in polarized abelian varieties.
In this paper we determine the structure of the Chow ring of the Delaunay-Voronoi compactification $\tilde{\cal A}_3$ of the moduli space of principally polarized abelian threefolds. This compactification was introduced by Namikawa and…
In this text we prove that if an abelian variety $A$ admits of an embedding into the Jacobian of a smooth projective curve $C$, and if we consider $\Th_A$ to be the divisor $\Th_C\cap A$, where $\Th_C$ denotes the theta divisor of $J(C)$,…
We study the conormal geometry theta divisors of certain singular bielliptic curves. We apply these results to the boundary components $\mathscr{S}_\underline{d}$ of the bielliptic Prym locus. We obtain results on the Gauss map, compute the…
The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of…
It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the…
We classify GV-subschemes of five-dimensional ppavs, showing that they exist only on Jacobians of curves and intermediate Jacobians of cubic threefolds, and confirming a conjecture of Pareschi--Popa in this case. The result is implied by a…
This survey examines separation of variables for algebraically integrable Hamiltonian systems whose tori are Jacobians of Riemann surfaces. For these cases there is a natural class of systems which admit separations in a nice geometric…
We prove a lower bound for the codimension of the Andreotti-Mayer locus N_{g,1} and show that the lower bound is reached only for the hyperelliptic locus in genus 4 and the Jacobian locus in genus 5. In relation with the boundary of the…
We compute the classes of universal theta divisors of degrees zero and g-1 over the Deligne-Mumford compactification of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of…
We show that a generic principally polarized abelian variety (ppav) is uniquely determined by its theta hyperplanes. These are the non-projectivized version of those studied by Caporaso and Sernesi (see math.AG/0204164), which in a sense…
We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X=V+W. For instance, we prove an optimal lower bound on the degree of the corresponding…