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On higher syzygies of ruled surfaces
Authors:Euisung Park
Affiliation: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 $ 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-aegeq 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-aegeq 3+p$.

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