Singularities of divisors of small degrees on abelian varieties

Abstract

Pareschi showed that an ample divisor of degree $d$ on a simple abelian variety of dimension $g$ has mild singularities when $d<g$. We extend his result to general polarized abelian varieties. We also show that ample divisors of degree $3$ or $4$ on an abelian variety have mild singularities.