# Singularities of divisors of low degree on simple abelian varieties

## TL;DR

This paper proves that divisors with low degree polarizations on simple abelian varieties have mild singularities, extending previous results and confirming a conjecture by Debarre and Hacon.

## Contribution

It extends known results on singularities of divisors to polarizations of degree less than the dimension on simple abelian varieties, settling a conjecture.

## Key findings

- Divisors of degree less than the dimension on simple abelian varieties have mild singularities.
- Confirmed a conjecture of Debarre and Hacon.
- Extended the class of polarizations for which singularity properties are understood.

## Abstract

It is known by results of Koll\'ar, Ein, Lazarsfeld, Hacon and Debarre that divisors representing principal and other low degree polarizations on abelian varieties have mild singularities. In this note we extend such results to polarizations of degree $<g$ on simple $g$-dimensional abelian varieties, settling a conjecture of Debarre and Hacon.