Higher syzygies of hyperelliptic curves (Q1039765)

Let \(K\) be an algebraically closed field of arbitrary characteristic and let \(\mathbb{P}^r\) be the projective \(r\)-space over \(K.\) In the theory of projective curves, one of the most important problems is to understand how a curve can map to projective spaces. Let \(L\) be a very ample line bundles of degree \(d\leq 2g\) on a hyperelliptic curve \(X\). Let \(\phi:X\to \mathbb{P}^r\) be the map induced by \(L.\) In this paper a couple of integers, the factorization type of \(L,\) is associated to \(L\) and it is shown that the Hartshorne-Rao module and the graded Betti numbers of the curve embedded by \(|L|\) are precisely determined by the factorization type.