On higher syzygies of ruled surfaces
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- by Euisung Park PDF
- Trans. Amer. Math. Soc. 358 (2006), 3733-3749 Request permission
Abstract:
We study higher syzygies of a ruled surface $X$ over a curve of genus $g$ with the numerical invariant $e$. Let $L \in \mbox {Pic}X$ 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 $b-ae\geq 3+p$.References
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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
- 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).
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 3733-3749
- MSC (2000): Primary 13D02, 14J26, 14N05
- DOI: https://doi.org/10.1090/S0002-9947-05-03875-4
- MathSciNet review: 2218997