On higher syzygies of ruled surfaces III

Abstract

In this paper, we study the minimal free resolution of homogeneous coordinate rings of a ruled surface S over a curve of genus g with the numerical invariant e < 0 and a minimal section C-0. Let L is an element of PicX be a line bundle in the numerical class of aC(0)+ b f such that a >= 1 and 2b ae = 4g 1 + k for some k >= max(2, e). We prove that the Green Lazarsfeld index index(S, L) of (S, L), i.e. the maximum p such that L satisfies condition N2,p, satisfies the inequalities k/2 - g <= k/2 - ae+3/2 + max (0, [2g-3+ae-k/4])) Also if S has an effective divisor D equivalent to 2C(0) + ef, then we obtain another upper bound of index(S, L), i.e., index(S, L) < k max(0, [2g-4-k/2. This gives a better bound in case b is small compared to a. Finally, for each e is an element of {g,, 1} we construct a ruled surface S with the numerical invariant e and a minimal section Co which has an effective divisor D equivalent to a-- 2C(0) + ef. (C) 2015 Elsevier B.V. All rights reserved.