Andreotti-Mayer loci and the Schottky problem

分支二重覆盖的普里姆轨迹与广义安德烈奥蒂-迈耶轨迹

Prym loci of branched double coverings and generalized Andreotti-Mayer loci

本文将安德烈奥蒂-迈耶轨迹推广至非主极化阿贝尔簇模空间，证明分支二重覆盖的普里姆轨迹是广义安德烈奥蒂-迈耶轨迹的不可约分支。

Principally polarized semi-abelic varieties of small torus rank, and the Andreotti-Mayer loci

We obtain, by a direct computation, explicit descriptions of all principally polarized semi-abelic varieties of torus rank up to 3. We describe the geometry of their symmetric theta divisors and obtain explicit formulas for the involution and its fixed points. These results allow us to give a new proof of the statements about the dimensions, for small genus, of the loci of ppav with theta divisor containing two-torsion points of multiplicity three. We also prove a result about the closure of this set. Our computations used in our work arXiv:1103.1857 to compute the class of the closure of the …

Semicontinuity of Gauss maps and the Schottky problem

We show that the degree of Gauss maps on abelian varieties is semicontinuous in families, and we study its jump loci. As an application we obtain that in the case of theta divisors this degree answers the Schottky problem. Our proof computes the degree of Gauss maps by specialization of Lagrangian cycles on the cotangent bundle. We also get similar results for the intersection cohomology of varieties with a finite morphism to an abelian variety; it follows that many components of Andreotti-Mayer loci, including the Schottky locus, are part of the stratification of the moduli space of ppav's de…

The Moduli Space of Abelian Varieties and the Singularities of the Theta Divisor

In this paper we give a lower bound for the codimension of the Andreotti-Mayer loci in the moduli space of principally polarized complex abelian varieties. We also present a conjecture on this codimension.

Jacobians with a vanishing theta-null in genus 4

In this paper we prove a conjecture of Hershel Farkas that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the abelian variety is a Jacobian. We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infinitesimal version of Andreotti and Mayer's local characterization of Jacobians by the dimension of the singular locus of the theta divisor.

具有横截 $\mathrm{A}_1$ 奇点的 Theta 除子上的 Gauss 映射

The Gauss Map on Theta Divisors with Transversal $\mathrm{A}_1$ Singularities

本文利用 Lagrangian specialization 计算具有横截 A1 奇点的 Theta 除子上 Gauss 映射的次数，并揭示其首系数与奇异轨迹 Samuel 重数的关系。