On the principally polarized abelian varieties that contain m-minimal curves

TL;DR

This paper investigates principally polarized abelian varieties containing curves that are multiples of the minimal class, generalizing Welters' work for m=2 to any m, and computes the dimension of the associated loci.

Contribution

It extends Welters' formalism to arbitrary m, constructing families of abelian varieties with m-minimal curves and determining the dimension of their moduli loci.

Findings

Constructed families of abelian varieties with m-minimal curves for any m.

Generalized Welters' formalism of stable pairs to broader cases.

Computed the dimension of the locus of such abelian varieties.

Abstract

In this paper, we study principally polarized abelian varieties $X$ of dimension $g$ that contain a curve $\nu:C\to X$ such that the class of $C$ is $m$ times the minimal class. Welters introduced the formalism of stable pairs to handle this problem in the case $m=2$. We generalize the results of Welters and construct families of principally polarized abelian varieties for any $m$ and compute the dimension of the locus of these abelian varieties.