The regularity of partial elimination ideals, Castelnuovo normality and syzygies

Abstract

Let X be a reduced closed subscheme in Pn, π:X→π(X)⊂Pn−1 be a projection from a point outside X and Zi(X)⊂π(X) be the closed subscheme defined by the i-th partial elimination ideal Ki(IX), which is supported on the (i+1)-th multiple points of π.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 Ki(IX),i≥1 (or that of the multiple locus Zi(X) of π). 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].