Frobenius split anticanonical divisors

TL;DR

This paper extends Sommese's theorems about abelian varieties to arbitrary characteristic, showing they cannot be ample divisors or smoothable cones over such varieties.

Contribution

It generalizes key properties of abelian varieties to all characteristics, broadening the scope of previous results.

Findings

Abelian varieties cannot be ample divisors in smooth projective varieties.

Cones over abelian varieties of dimension at least two are not smoothable.

Abstract

In this note I extend two theorems of Sommese regarding abelian varieties to arbitrary characteristic; that an abelian variety cannot be an ample divisor in a smooth projective variety and that a cone over an abelian variety of dimension at least two is not smoothable.