REMARKS ON SYZYGIES OF THE SECTION MODULES AND GEOMETRY OF PROJECTIVE VARIETIES

Abstract

Let X subset of 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 R(L) = circle plus(l is an element of Z) H(0)(X, L(l)) 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.