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Transactions of the American Mathematical Society

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On higher syzygies of ruled surfaces


Author: Euisung Park
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
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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  • 1. David C. Butler, Normal generation of vector bundles over a curve, J. Differ. Geom. 39 (1994), 1-34. MR 1258911 (94k:14024)
  • 2. David Eisenbud, The Geometry of Syzygies, Graduate Texts in Math., no. 229, Springer-Verlag, New York (2005). MR 2103875 (2005h:13021)
  • 3. F. J. Gallego and B. P. Purnaprajna, Normal Presentation on Elliptic Ruled Surfaces, J. Algebra 186 (1996), 597-625. MR 1423277 (98c:14030)
  • 4. F. J. Gallego and B. P. Purnaprajna, Higher syzygies of elliptic ruled surfaces, J. Algebra 186 (1996), 626-659. MR 1423278 (98c:14031)
  • 5. F. J. Gallego and B. P. Purnaprajna, Projective normality and syzygies of algebraic surfaces, J. Reine Angew. Math. 506 (1999), 145-180. MR 1665689 (2000a:13023)
  • 6. F. J. Gallego and B. P. Purnaprajna, Vanishing theorems and syzygies for K3 surfaces and Fano varieties, J. Pure App. Alg. 146 (2000), 251-265. MR 1742342 (2001f:14071)
  • 7. F. J. Gallego and B. P. Purnaprajna, Some results on rational surfaces and Fano varieties, J. Reine Angew. Math. 538 (2001), 25-55. MR 1855753 (2002f:14024)
  • 8. M. Green, Koszul cohomology and the geometry of projective varieties I, J. Differ. Geom. 19 (1984), 125-171. MR 0739785 (85e:14022)
  • 9. M. Green and R. Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Inv. Math. 83 (1986), 73-90. MR 0813583 (87g:14022)
  • 10. M. Green and R. Lazarsfeld, Some results on the syzygies of finite sets and algebraic curves, Compositio Mathematica 67 (1988), 301-314. MR 0959214 (90d:14034)
  • 11. Robin Hartshorne, Algebraic Geometry, no. 52, Springer-Verlag, New York (1977). MR 0463157 (57:3116)
  • 12. Y. Homma, Projective normality and the defining equations of ample invertible sheaves on elliptic ruled surfaces with $ e \geq 0$, Natural Sci. Rep. Ochanomizu Univ. 31 (1980), 61-73. MR 0610593 (82e:14044)
  • 13. Y. Homma, Projective normality and the defining equations of an elliptic ruled surface with negative invariant, Natural Sci. Rep. Ochanomizu Univ. 33 (1982), 17-26. MR 0703959 (85a:14027)
  • 14. Euisung Park, On higher syzygies of ruled varieties over a curve, to appear in J. Algebra.

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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: https://doi.org/10.1090/S0002-9947-05-03875-4
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

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