Regularity of structure sheaves of varieties with isolated singularities

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.

A mistake in Section 3.4 of the last version is corrected with the proof presented in Section 4 of the current version. Previous Section 4, now 5, is simplified