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 − 2, L satisfies Property Np if a ≥ p + 2 and b − ae ≥ 3g − e + p and for e ≥ g − 1, L satisfies Property Np if a ≥ p + 2 and b − ae ≥ 2g + 1 + p. By using these facts, we obtain Mukai type results. For ample line bundles Ai, we show that KX +A1+ · · ·+Aq satisfies Property Np when 0 ≤ e < g−2 2 and q ≥ g − 2e + 2 + p or when e ≥ g−2 2 and q ≥ p + 4. Therefore we prove Mukai’s conjecture for ruled surface with e ≥ g−2 2 . Also we prove that when X is an elliptic ruled surface with e ≥ 0, L satisfies Property Np if and only if a ≥ 1 and b − ae ≥ 3 + p.