Higher syzygies of hyperelliptic curves

Let X be a hyperelliptic curve of arithmetic genus g and let f : X → P¹ be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d ≤ 2 g. Note that the minimal free resolution of X of degree ≥ 2 g + 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 ≤ 2 g. For m = max {t ∈ Z {divides} H⁰ (X, L ⊗ A^{- t}) ≠ 0}, we call the pair (m, d - 2 m)the factorization type ofL . Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by | L | are precisely determined by the factorization type of L.