The loci of abelian varieties with points of high multiplicity on the theta divisor

TL;DR

This paper investigates the geometric loci of abelian varieties with high multiplicity points on their theta divisors, using advanced techniques to relate their structures and establish bounds on their dimensions.

Contribution

It introduces new bounds and relations for the dimensions of these loci, and proposes conjectures about their geometric structure.

Findings

Bounds on the dimensions of loci with high multiplicity points

Relations among the dimensions of different loci

Conjectures on the structure of these loci

Abstract

We study the loci of principally polarized abelian varieties with points of high multiplicity on the theta divisor. Using the heat equation and degeneration techniques, we relate these loci and their closures to each other, as well as to the singular set of the universal theta divisor. We obtain bounds on the dimensions of these loci and relations among their dimensions, and make further conjectures about their structure.