On higher syzygies of ruled surfaces

Abstract

We study higher syzygies of a ruled surface $X$ over a curve of genus $g$ with the numerical invariant $e$. Let $L\in PicX$ be a line bundle in the numerical class of $a{C}_0+bf$. We prove that for $0\le e\le g-3$, $L$ satisfies property $N_p$ if $a\ge p+2$ and $b-ae\ge 3g-1-e+p$ and for $e\ge g-2$, $L$ satisfies property $N_p$ if $a\ge p+2$ and $b-ae\ge 2g+1+p$. By using these facts, we obtain Mukai type results. For ample line bundles $A_i$, we show that $K_X+A_1+...+A_q$ satisfies property $N_p$ when $0\le e<\frac{g-3}{2}$ and $q\ge g-2e+1+p$ or when $e\ge \frac{g-3}{2}$ and $q\ge p+4$. Therefore we prove Mukai's conjecture for ruled surface with $e\ge \frac{g-3}{2}$. Also we prove that when $X$ is an elliptic ruled surface with $e\ge 0$, $L$ satisfies property $N_p$ if and only if $a\ge 1$ and $b-ae\ge 3+p$.

19 pages, 3 figures