On higher syzygies of ruled surfaces III

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<0e<0 and a minimal section C0C0. Let L∈PicXL∈PicX be a line bundle in the numerical class of aC0+bfaC0+bf such that a≥1a≥1 and 2b−ae=4g−1+k2b−ae=4g−1+k for some k≥max(2,−e)k≥max(2,−e). We prove that the Green–Lazarsfeld index index(S,L)index(S,L) of (S,L)(S,L), i.e. the maximum p such that L   satisfies condition N2,pN2,p, satisfies the inequalitiesk2−g≤index(S,L)≤k2−ae+32+max(0,⌈2g−3+ae−k4⌉). Also if S   has an effective divisor D≡2C0+efD≡2C0+ef, then we obtain another upper bound of index(S,L)index(S,L), i.e., index(S,L)≤k+max(0,⌈2g−4−k2⌉). This gives a better bound in case b is small compared to a  . Finally, for each e∈{−g,…,−1}e∈{−g,…,−1} we construct a ruled surface S with the numerical invariant e   and a minimal section C0C0 which has an effective divisor D≡2C0+efD≡2C0+ef.