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 $X={\mathrm{\P }}_C(\mathcal{E})$ 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)$bethetautologicallinebundle,weprovethatif$(\P^n,$\mathcal{O}${\P^n} (a))$satisfiesProperty$N_p$,then$(X,aH+\pi^*B)$satisfiesProperty$N_p$forall$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 $rank(\mathcal{E})\ge g$ 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 $A\in PicX$ which shows that the condition of Mukai’s conjecture is optimal for every $p\ge 0$.