On syzygies of Veronese embedding of arbitrary projective varieties

Abstract

For a very ample line bundle L on a projective scheme X, let phi(Ll) : X hooked right arrow PH(0) (X, L(l)), l >= 1, be the embedding defined by the complete linear series |L(l)|. In this paper we study the problem how the Castelnuovo-Mumford regularity of phi(L)(X) effects on the defining equations of phi(Ll)(X) and the syzygies among them. We show that if phi(L)(X) subset of PH(0) (X, L) is m-regular, then (X, L(l)) satisfies property N(2l-m+1) for m/2 <= l <= m - 2, and (X, L(l)) satisfies property N(l) for all l >= m - 1. (C) 2009 Elsevier Inc. All rights reserved.