On higher syzygies of ruled surfaces

We study higher syzygies of a ruled surface X over a curve of genus g with the numerical invariant e. Let L ∈ PicX be a line bundle in the numerical class of aC₀ + bf. We prove that for 0 ≤ e ≤ g - 3, L satisfies property N_pif a ≥ p+2 and b - ae ≥ 3g - 1 - e+p, and for e ≥ g - 2, L satisfies property N_p if a ≥ p+2 and b - ae ≥ 2g + 1+ p. By using these facts, we obtain Mukai-type results. For ample line bundles A_i, we show that K_X+A₁+⋯+A _q satisfies property N_p when 0 ≤ e < g-3/2 and q ≥ g -2e+1+p or when e ≥ g-3/2 and q ≥ p + 4. Therefore we prove Mukai's conjecture for ruled surface with e ≥ g-3/2. We also prove that when X is an elliptic ruled surface with e ≥ 0, L satisfies property N_p if and only if a ≥ 1 and b - ae ≥ 3 + p.