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

TL;DR

This paper investigates the singularities of pluri-theta divisors on ppav in characteristic p>0, showing they are limits of strongly F-regular pairs and establishing bounds on multiplicities.

Contribution

It demonstrates that pluri-theta divisors on ppav in characteristic p>0 are limits of strongly F-regular pairs and provides bounds on their multiplicities.

Findings

$(X,rac{1}{m}D)$ is a limit of strongly F-regular pairs

${ m mult}_x(D) leq m ext{dim }X$ for all $x",

Abstract

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 $x\in X$.