Higher syzygies of hyperelliptic curves

Let XX be a hyperelliptic curve of arithmetic genus gg and let f:X→P1f:X→P1 be the hyperelliptic involution map of XX. In this paper we study higher syzygies of linearly normal embeddings of XX of degree d≤2gd≤2g. Note that the minimal free resolution of XX of degree ≥2g+1≥2g+1 is already completely known. Let A=f∗OP1(1)A=f∗OP1(1), and let LL be a very ample line bundle on XX of degree d≤2gd≤2g. For m=max{t∈Z∣H0(X,L⊗A−t)≠0}, we call the pair (m,d−2m)(m,d−2m)the factorization type of  LL. Our main result is that the Hartshorne–Rao module and the graded Betti numbers of the linearly normal curve embedded by |L||L| are precisely determined by the factorization type of LL.