The regularity of partial elimination ideals, Castelnuovo normality and syzygies

Abstract

Let X be a reduced closed subscheme in P-n, pi : X -> pi(X) subset of Pn-1 be a projection from a point outside X and Z(i) (X) subset of pi(X) be the closed subscheme defined by the i-th partial elimination ideal K-i(I-x), which is supported on the (i + 1)-th multiple points of pi. In this paper, motivated from projection methods to prove Eisenbud-Goto conjecture on regularity in many cases, we describe the syzygetic behaviors and Castelnuovo normality of the projection with a viewpoint of the regularity of the partial elimination ideal K-i(I-X), i >= 1 (or that of the multiple locus Z(i) (X) of pi). We also give some applications to the syzygies and Castelnuovo normality of successive projections, which recover and generalize some known results in [1,3,15,16]. (C) 2019 Published by Elsevier Inc.