Abstract | 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 ∈ Pic X be a line bundle in the numerical class of a C 0 + b f such that a ≥ 1 and 2 b − a e = 4 g − 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 inequalities k/2 − g ≤ index ( S , L ) ≤ k/2 − (a e + 3)/2 + max ( 0 , ⌈ (2 g − 3 + a e − k)/4 ⌉ ) . Also if S has an effective divisor D ≡ 2 C 0 + e f , then we obtain another upper bound of index ( S , L ) , i.e., index ( S , L ) ≤ k + max ( 0 , ⌈ (2 g − 4 − k)/2 ⌉ ) . 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 C 0 which has an effective divisor D ≡ 2 C 0 + e f . |