Singularities of divisors of small degrees on abelian varieties

Pareschi showed that an ample divisor of degree d on a simple abelian variety of dimension g has mild singularities when \(d<g\) $d<g$ . We extend his result to general polarized abelian varieties. We also show that ample divisors of degree 3 (resp. 4) on an abelian variety of dimension \(\ge 3\) $\ge 3$ (resp. \(\ge 4\) $\ge 4$ ) have mild singularities, which extends Debarre and Hacon’s previous work on polarizations of degree 2.