On higher syzygies of ruled surfaces II

Abstract

In this article we continue the study of property $N_p$ of irrational ruled surfaces begun in [E. Park, On higher syzygies of ruled surfaces, math.AG/0401100, Trans. Amer. Math. Soc., in press]. Let X be a ruled surface over a curve of genus $g⩾1$ with a minimal section $C_0$ and the numerical invariant e. When X is an elliptic ruled surface with $e=-1$, it is shown in [F.J. Gallego, B.P. Purnaprajna, Higher syzygies of elliptic ruled surfaces, J. Algebra 186 (1996) 626–659] that there is a smooth elliptic curve $E\subset X$ such that $E\equiv 2C_0-f$. And we prove that if $L\in \mathrm{Pic}X$ is in the numerical class of $a{C}_0+bf$ and satisfies property $N_p$, then $(C,L{|}_{C_0})$ and $(E,L{|}_E)$ satisfy property $N_p$ and hence $a+b⩾3+p$ and $a+2b⩾3+p$. This gives a proof of the relevant part of Gallego–Purnaprajna' conjecture in [F.J. Gallego, B.P. Purnaprajna, Higher syzygies of elliptic ruled surfaces, J. Algebra 186 (1996) 626–659]. When $g⩾2$ and $e⩾0$ we prove some effective results about property $N_p$. Let $L\in \mathrm{Pic}X$ be a line bundle in the numerical class of $a{C}_0+bf$. 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 $e⩾g-2$ and $b-ae⩾3g-2$, then L satisfies property $N_p$ if and only if $b-ae⩾2g+1+p$. (2) When C is a hyperelliptic curve of genus $g⩾2$, L is normally generated if and only if $b-ae⩾2g+1$ and normally presented if and only if $b-ae⩾2g+2$. Also if $e⩾g-2$, then L satisfies property $N_p$ if and only if $a⩾1$ and $b-ae⩾2g+1+p$.