2022/03/09 by Wenbo Niu, Jinhyung Park, Niu, Wenbo +1
Arts and Humanities · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #North African History and Literature
paper · pdf · doi:10.48550/arxiv.2203.04601
openalex publication_date 2022/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.