Remarks on Syzygies of the Section Modules and Geometry of Projective Varieties

Abstract

Let X ⊂ ℙ(H 0(ℒ)) be a smooth projective variety embedded by the complete linear system associated to a very ample line bundle ℒ on X. We call the section module of ℒ. It has been known that the syzygies of R ℒ as R = Sym(H 0(ℒ))-module play important roles in understanding geometric properties of X [2, 3, 5, 9, 10] even if X is not projectively normal. Generalizing the case of N 2, p [2, 10], we prove some uniform theorems on higher normality and syzygies of a given linearly normal variety X and general inner projections when R ℒ satisfies property N 3, p (Theorems 1.1, 1.2, and Proposition 3.1). In particular, our uniform bounds are sharp as hyperelliptic curves and elementary transforms of elliptic ruled surfaces show.