Theta-regularity and log-canonical threshold

TL;DR

This paper explores the relationship between log-canonical thresholds and theta-regularity, extending known inequalities from projective space to principally polarized abelian varieties using advanced regularity concepts.

Contribution

It generalizes an inequality relating log-canonical thresholds and Castelnuovo-Mumford regularity to the setting of abelian varieties via theta-regularity.

Findings

Established a new inequality involving theta-regularity and log-canonical thresholds.

Extended the applicability of known bounds from projective space to abelian varieties.

Provided a framework for analyzing singularities in abelian varieties using theta-regularity.

Abstract

We show that an inequality, proven by K\"uronya-Pintye, which governs the behavior of the log-canonical threshold of an ideal over $\mathbb{P}^n$ and that of its Castelnuovo-Mumford regularity, can be applied to the setting of principally polarized abelian varieties by substituting the Castelnuovo-Mumford regularity with $\Theta$-regularity of Pareschi-Popa.