Regularity of structure sheaves of varieties with isolated singularities

TL;DR

This paper proves a bound on the regularity of structure sheaves for certain projective varieties with isolated singularities, confirming a conjecture in this specific setting.

Contribution

It establishes the Castelnuovo-Mumford regularity bound for varieties with isolated $ ext{Q}$-Gorenstein singularities, using classification and vanishing techniques.

Findings

Regularity bound $ ext{reg}( ext{O}_X) extless= d - e

Classification of extremal and near-extremal cases

Bound fails for general projective varieties

Abstract

Let $X\subseteq \mathbb{P}^N$ be a non-degenerate normal projective variety of codimension $e$ and degree $d$ with isolated $\mathbb{Q}$-Gorenstein singularities. We prove that the Castelnuovo-Mumford regularity $\text{reg}(\mathcal{O}_{X})\le d-e$, as predicted by the Eisenbud-Goto regularity conjecture. Such a bound fails for general projective varieties. The main techniques are Noma's classification of non-degenerated projective varieties and Nadel vanishing for multiplier ideals. We also classify the extremal and the next to extremal cases.