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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On higher syzygies of ruled surfaces


Author: Euisung Park
Journal: Trans. Amer. Math. Soc. 358 (2006), 3733-3749
MSC (2000): Primary 13D02, 14J26, 14N05
Published electronically: December 27, 2005
MathSciNet review: 2218997
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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-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 + \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-ae\geq 3+p$.


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Additional Information

Euisung Park
Affiliation: School of Mathematics, Korea Institute for Advanced Study, 207-43 Cheongryangri 2-dong, Dongdaemun-gu, Seoul 130-722, Republic of Korea
Email: puserdos@kias.re.kr

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03875-4
PII: S 0002-9947(05)03875-4
Received by editor(s): January 26, 2004
Received by editor(s) in revised form: October 16, 2004
Published electronically: December 27, 2005
Additional Notes: The author was supported by Korea Research Foundation Grant (KRF-2002-070-C00003).
Article copyright: © Copyright 2005 American Mathematical Society