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<0 and a minimal section C₀. Let L∈PicX be a line bundle in the numerical class of aC₀+bf 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 N_2,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+ef, then we obtain another upper bound of 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} we construct a ruled surface S with the numerical invariant e and a minimal section C₀ which has an effective divisor D≡2C0+ef.