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On higher syzygies of ruled surfaces

Authors:	Euisung Park

Institution:	School of Mathematics, Korea Institute for Advanced Study, 207-43 Cheongryangri 2-dong, Dongdaemun-gu, Seoul 130-722, Republic of Korea

Abstract:	We study higher syzygies of a ruled surface over a curve of genus with the numerical invariant . Let $L \in$ Pic be a line bundle in the numerical class of . We prove that for $0 \leq e \leq g-3$ , satisfies property if $a \geq p+2$ and $b-ae \geq 3g-1-e+p$ , and for $e \geq g-2$ , satisfies property 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 , we show that $K_X + A_1 + \cdots + A_q$ satisfies property 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 is an elliptic ruled surface with $e \geq 0$ , satisfies property if and only if $a \geq 1$ and $b-ae\geq 3+p$ .

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