A Castelnuovo-Mumford regularity bound for threefolds with rational singularities

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 Pr of degree d and codimension e. We prove that if X has rational singularities, then reg(X)≤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 reg(X)≤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.