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

TL;DR

This paper explicitly describes small torus rank principally polarized semi-abelic varieties, analyzing their theta divisors and involutions, providing new proofs and insights into the geometry of Andreotti-Mayer loci.

Contribution

It offers explicit geometric descriptions and formulas for semi-abelic varieties of small torus rank, advancing understanding of Andreotti-Mayer loci and their closures.

Findings

Explicit descriptions of semi-abelic varieties of torus rank up to 3.

Formulas for involutions and fixed points on theta divisors.

New proofs of dimension statements for small genus ppav loci.

Abstract

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 locus of intermediate jacobians of cubic threefolds in the Chow ring of the perfect cone compactification of the moduli space of principally polarized abelian fivefolds.