Higher syzygies of hyperelliptic curves

Abstract

Let X be a hyperelliptic curve of arithmetic genus g and let f : X -> P-1 be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d <= 2g. Note that the minimal free resolution of X of degree >= 2g + 1 is already completely known. Let A = f*O-P1(1), and let L be a very ample line bundle on X of degree d <= 2g. For m = max {t is an element of Z} H-0(X, L circle times A(-t)) not equal 0}, we call the pair (m, d-2m) the factorization type of L. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by vertical bar L vertical bar are precisely determined by the factorization type of L. (C) 2009 Elsevier B.V. All rights reserved.