Higher syzygies of ruled varieties over a curve

Let X be a ruled variety over a smooth projective curve C with the projection morphism . In this paper we study higher syzygies of very ample line bundles on X.Each embedding of X is fiberwise a Veronese embedding. And our first result is to clarify the relation between property Np of very ample line bundles on X and that of the Veronese embedding. More precisely, letting H be the tautological line bundle of X, assume that the a-uple Veronese embedding of a fiber satisfies property Np. We prove that line bundles on X of the form aH+π∗B satisfy property Np if deg(B) is sufficiently large (Theorem 1.1). Also we get some partial answer for the converse (Corollary 3.7). From this observation, we improve Butler's result in [D.C. Butler, Normal generation of vector bundles over a curve, J. Differential Geom. 39 (1994) 1–34] for ruled scrolls, ruled surfaces and Veronese surface fibrations.