On higher syzygies of ruled surfaces

We study higher syzygies of a ruled surface $X$ over a curve of genus $g$ with the numerical invariant $e$. Let $L \in$   Pic$X$ be a line bundle in the numerical class of $aC_0 +bf$. We prove that for $0 \leq e \leq g-3$, $L$ satisfies property $N_p$ if $a \geq p+2$ and $b-ae \geq 3g-1-e+p$, and for $e \geq g-2$, $L$ satisfies property $N_p$ if $a \geq p+2$ and $b-ae\geq 2g+1+p$. By using these facts, we obtain Mukai-type results. For ample line bundles $A_i$, we show that $K_X + A_1 + \cdots + A_q$ satisfies property $N_p$ when $0 \leq e < \frac{g-3}{2}$ and $q \geq g-2e+1 +p$ or when $e \geq \frac{g-3}{2}$ and $q \geq p+4$. Therefore we prove Mukai's conjecture for ruled surface with $e \geq \frac{g-3}{2}$. We also prove that when $X$ is an elliptic ruled surface with $e \geq 0$, $L$ satisfies property $N_p$ if and only if $a \geq 1$ and $b-ae\geq 3+p$.