Complete Linear Series on a Hyperelliptic Curve

In this paper we study complete linear series on a hyperelliptic curve C of arithmetic genus g. Let A be the unique line bundle on C such that |A| is a g 2 , and let L be a line bundle on C of degree d. Then L can be factorized as L = A ⊗ B where m is the largest integer satisfying H(C,L⊗A) 6= 0. Let b = deg(B). We say that the factorization type of L is (m, b). Our main results in this paper assert that (m, b) gives a precise answer for many natural questions about L. We first show that (m, b) precisely determines the dimension of the vector spaces H(C,L) and H(C,L), and the base point freeness and the very ampleness of L. For example, L is very ample if and only if b = 0 and m ≥ g + 1 or 1 ≤ b ≤ g + 1 and m + b ≥ g + 2. When L is very ample, we study the Hartshorne-Rao module and the minimal free resolution of the linearly normal curve embedded by |L|. For d = 2g + 1 + p, p ≥ 0, we obtain all the graded Betti numbers explicitly. In this case, property Np holds while property Np+1 fails to hold. We show that a finite subscheme of C determined by the factorization of L causes the failure of Np+1. For d ≤ 2g, we discuss at length the Hartshorne-Rao module and the minimal free resolution. It turns out that they are precisely determined by (m, b). In particular, it is shown that the two line bundles have the same factorization type if and only if the Betti diagrams of the corresponding linearly normal curves are equal to each other. This enables us to understand how many distinct Betti diagrams occur at all.