On Higher Syzygies of ruled varieties over a curve

For a vector bundle $\mathcal{E}$ of rank $n+1$ over a smooth projective curve $C$ of genus $g$, let $X=\P_C (\mathcal{E})$ with projection map $\pi:X\to C$. 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 $\pi$ by the same line bundle. More precisely, letting $H = \mathcal{O}_{\P_C (\mathcal{E})} (1)$ be the tautological line bundle, we prove that if $(\P^n,\mathcal{O}_{\P^n} (a))$ satisfies Property $N_p$, then $(X,aH+\pi^*B)$ satisfies Property $N_p$ for all $B \in {Pic}C$ having sufficiently large degree(Theorem \ref{thm:positive}). And also the effective bound of ${deg}(B)$ for Property $N_p$ is obtained(Theorem \ref{thm:1}, \ref{thm:2}, \ref{thm:3} and \ref{thm:4}). For the converse, we get some partial answer(Corollary \ref{cor:negative}). Secondly, by using these results we prove some Mukai-type statements. In particular, Mukai's conjecture is true for $X$ when ${rank}(\mathcal{E}) \geq g$ and $\mu^- (\mathcal{E})$ is an integer(Corollary \ref{cor:Mukai}). Finally for all $n$, we construct an $n$-dimensional ruled variety $X$ and an ample line bundle $A \in {Pic}X$ which shows that the condition of Mukai's conjecture is optimal for every $p \geq 0$.