On higher syzygies of ruled surfaces III

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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 N2,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.