A Castelnuovo-Mumford regularity bound for threefolds with rational singularities

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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 P-r 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. (C) 2022 Elsevier Inc. All rights reserved.
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Issue Date
2022-06
Language
English
Article Type
Article
Citation

ADVANCES IN MATHEMATICS, v.401

ISSN
0001-8708
DOI
10.1016/j.aim.2022.108320
URI
http://hdl.handle.net/10203/296706
Appears in Collection
MA-Journal Papers(저널논문)
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