Singularities of pluri-theta divisors in Char $p > 0$

Abstract

We show that if $(X, \Theta)$ is a PPAV over an algebraically closed field of characteristic $p \gt 0$ and $D \in |m\Theta |$, then $(X, \frac 1m D)$ is a limit of strongly $F$-regular pairs and in particular $\mathrm{mult}_x (D) \leq m \cdot \dim X$ for any $x \in X$.