On higher syzygies of ruled surfaces II

Abstract

In this article we we continue the study of property $N_p$ of irrational ruled surfaces begun in \cite{ES}. Let $X$ be a ruled surface over a curve of genus $g \geq 1$ with a minimal section $C_0$ and the numerical invariant $e$. When $X$ is an elliptic ruled surface with $e = -1$, there is an elliptic curve $E \subset X$ such that $E \equiv 2C_0 -f$. And we prove that if $L \in {Pic}X$ is in the numerical class of $aC_0 +bf$ and satisfies property $N_p$, then $(C,L|_{C_0})$ and $(E,L|_E)$ satisfy property $N_p$ and hence $a+b \geq 3+p$ and $a+2b \geq 3+p$. This gives a proof of the relevant part of Gallego-Purnaprajna' conjecture in \cite{GP2}. When $g \geq 2$ and $e \geq 0$ we prove some effective results about property $N_p$. Let $L \in {Pic}X$ be a line bundle in the numerical class of $aC_0 +bf$. Our main result is about the relation between higher syzygies of $(X,L)$ and those of $(C,L_{C})$ where $L_C$ is the restriction of $L$ to $C_0$. In particular, we show the followings: $(1)$ If $e \geq g-2$ and $b-ae \geq 3g-2$, then $L$ satisfies property $N_p$ if and only if $b-ae \geq 2g+1+p$. $(2)$ When $C$ is a hyperelliptic curve of genus $g \geq 2$, $L$ is normally generated if and only if $b-ae \geq 2g+1$ and normally presented if and only if $b-ae \geq 2g+2$. Also if $e \geq g-2$, then $L$ satisfies property $N_p$ if and only if $a \geq 1$ and $b-ae \geq 2g+1+p$.