A Castelnuovo-Mumford regularity bound for threefolds with rational singularities

TL;DR

This paper establishes a near-sharp bound on the Castelnuovo-Mumford regularity for threefolds with rational singularities, advancing understanding of algebraic geometry bounds for singular varieties.

Contribution

It provides a new regularity bound for threefolds with rational singularities and classifies extremal cases when certain conditions are met.

Findings

Regularity bound: reg(X) ≤ d - e + 2 for threefolds with rational singularities.

Special case: reg(X) ≤ d - 1 when e=2 and X has Cohen-Macaulay Du Bois singularities.

Method: Bounded regularity of fibers via Loewy length and secant line analysis.

Abstract

The purpose of this paper is to establish a Castelnuovo-Mumford regularity bound for threefolds with mild singularities. Let $X$ be a non-degenerate normal projective threefold in $\mathbb{P}^r$ of degree $d$ and codimension $e$. We prove that if $X$ has rational singularities, then $\text{reg}(X) \leq d-e+2$. Our bound is very close to a sharp bound conjectured by Eisenbud-Goto. When $e=2$ and $X$ has Cohen-Macaulay Du Bois singularities, we obtain the conjectured bound $\text{reg}(X) \leq d-1$, and we also classify the extremal cases. To achieve these results, we bound the regularity of fibers of a generic projection of $X$ by using Loewy length, and also bound the dimension of the varieties swept out by secant lines through the singular locus of $X$.