On Higher Syzygies of Ruled Varieties over a Curve

For a vector bundle E of rank n + 1 over a smooth projective curve C of genus g, let X = PC(E) with projection map π : X → C. In this paper we investigate the minimal free resolution of homogeneous coordinate rings of X . We first clarify the relations between higher syzygies of very ample line bundles on X and higher syzygies of Veronese embedding of fibres of π by the same line bundle. More precisely, letting H = OPC(E)(1) be the tautological line bundle, we prove that if (P,OPn(a)) satisfies Property Np, then (X, aH + π ∗B) satisfies Property Np for all B ∈ PicC having sufficiently large degree(Theorem 1.1). And also the effective bound of deg(B) for Property Np is obtained(Theorem 4.2, 4.5, 4.7 and 4.9). For the converse, we get some partial answer(Corollary 3.3). Secondly, by using these results we prove some Mukai-type statements. In particular, Mukai’s conjecture is true for X when rank(E) ≥ g and μ−(E) is an integer(Corollary 4.11). Finally for all n, we construct an n-dimensional ruled variety X and an ample line bundle A ∈ PicX which shows that the condition of Mukai’s conjecture is optimal for every p ≥ 0.