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

Singularities of pluri-theta divisors in Char $p &gt; 0$

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