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 $L\in PicX$ 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 $e \geq g-2$,$L$satisfiesproperty$N_p$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 $K_X+A_1+\mathrm{.}\mathrm{.}\mathrm{.}+A_q$ satisfies property $N_p$ when $0 \leq e < \frac{g-3}{2}$and$q \geq g-2e+1 +p$orwhen$e \geq \frac{g-3}{2}$and$q \geq p+4$\mathrm{.}ThereforeweproveMuka{i}^{\mathrm{\prime }}sconjectureforruledsurfacewith$e \geq \frac{g-3}{2}$\mathrm{.}Alsoweprovethatwhen$X$ is an elliptic ruled surface with $e\ge 0$, $L$ satisfies property $N_p$ if and only if $a\ge 1$ and $b-ae\ge 3+p$.