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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d<g$$\end{document}. 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 >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ge 3$$\end{document} (resp. >= 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ge 4$$\end{document}) have mild singularities, which extends Debarre and Hacon's previous work on polarizations of degree 2.