Singularities of base loci on abelian varieties

TL;DR

This paper establishes a lower bound for the log canonical threshold of base ideals on complex abelian varieties, characterizing when equality occurs based on the base locus structure.

Contribution

It proves that the log canonical threshold of the base ideal is at least 1, with equality precisely when the base locus contains divisorial components.

Findings

Log canonical threshold of base ideals is ≥ 1.

Equality occurs if and only if the base locus has divisorial components.

Results extend to ideals of intersections of translated theta divisors.

Abstract

We prove that the log canonical threshold of the base ideal of a complete linear system on a complex abelian variety is $\ge 1$, and equality holds if and only if the base locus has divisorial components. Consequently the same assertions hold for the ideal of the intersection of translates of theta divisors by the points of a finite subgroup.