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Green’s Conjecture for curves on rational surfaces with an anticanonical pencil

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Abstract

Green’s Conjecture is proved for smooth curves \(C\) lying on a rational surface \(S\) with an anticanonical pencil, under some mild hypotheses on the line bundle \(L=\mathcal{O }_S(C)\). Constancy of Clifford dimension, Clifford index and gonality of curves in the linear system \(\vert L\vert \) is also obtained.

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References

  1. Aprodu, M.: On the vanishing of higher syzygies of curves. Math. Zeit. 241, 1–15 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aprodu, M.: Remarks on syzygies of d-gonal curves. Math. Res. Lett. 12, 387–400 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aprodu, M.: Lazarsfeld-Mukai bundles and applications. ArXiv:1205.4415

  4. Aprodu, M., Farkas, G.: The Green conjecture for smooth curves lying on arbitrary \(K3\) surfaces. Compos. Math. 147, 839–851 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arbarello, E., Cornalba, M., Griffiths, P.A.: Geometry of algebraic curves volume II with a contribution by Joseph Daniel Harris, Grundlehren der mathematischen Wissenschaften, vol. 267. Springer, Berlin (2011)

    Google Scholar 

  6. Coppens, M., Martens, G.: Secant spaces and Clifford’s Theorem. Compos. Math. 78, 193–212 (1991)

    MathSciNet  MATH  Google Scholar 

  7. Costa, L., Miró-Roig, R.M.: Rationality of moduli spaces of vector bundles on rational surfaces. Nagoya Math. J. 165, 43–69 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Ehbauer, S.: Syzygies of points in projective space and applications. Zero-dimensional schemes (Ravello, (1992)), pp. 145–170. De Gruyter, Berlin (1994)

  9. Eisenbud, D., Lange, H., Martens, G., Schreyer, F.-O.: The Clifford dimension of a projective curve. Compos. Math. 72, 173–204 (1989)

    MathSciNet  MATH  Google Scholar 

  10. Green, M.L.: Koszul cohomology and the geometry of projective varieties. J. Diff. Geom. 19, 125–171 (1984)

    MATH  Google Scholar 

  11. Harbourne, B.: Birational morphisms of rational surfaces. J. Algebra 190, 145–162 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hirschowitz, A., Ramanan, S.: New evidence for Green’s conjecture on syzygies of canonical curves. Ann. Sci. Éc. Norm. Supér. (4) 31, 145–152 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Knutsen, A.L.: Exceptional curves on Del Pezzo surfaces. Math. Nachr. 256, 58–81 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lazarsfeld, R.: Brill-Noether-Petri without degenerations. J. Differ. Geom. 23, 299–307 (1986)

    MathSciNet  MATH  Google Scholar 

  15. Lazarsfeld, R.: Positivity in algebraic geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, vol. 49. Springer, Berlin (2004)

    Book  Google Scholar 

  16. Lelli-Chiesa, M.: Stability of rank-3 Lazarsfeld-Mukai bundles on \(K3\) surfaces. ArXiv:1112.2938

  17. Loose, F.: On the graded Betti numbers of plane algebraic curves. Manuscr. Math. 64, 503–514 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Martens, G.: Über den Clifford-Index algebraischer Kurven. J. Reine Angew. Math. 336, 83–90 (1982)

    MathSciNet  MATH  Google Scholar 

  19. Pareschi, G.: Exceptional linear systems on curves on Del Pezzo surfaces. Math. Ann. 291, 17–38 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pareschi, G.: A proof of Lazarsfeld’s Theorem on curves on \(K3\) surfaces. J. Alg. Geom. 4, 195–200 (1995)

    MathSciNet  MATH  Google Scholar 

  21. Voisin, C.: Green’s generic syzygy conjecture for curves of even genus lying on a \(K3\) surface. J. Eur. Math. Soc. 4, 363–404 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Voisin, C.: Green’s canonical syzygy conjecture for generic curves of odd genus. Compos. Math. 141, 1163–1190 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

I am grateful to my advisor Gavril Farkas, who suggested me to investigate the topic. This paper has been written during my Ph.D. studies in Berlin and I warmly thank the Berlin Mathematical School (BMS) for supporting me.

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  1. Max Planck Institute for Mathematics, 53111 , Bonn, Germany

    Margherita Lelli-Chiesa

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  1. Margherita Lelli-Chiesa
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Correspondence to Margherita Lelli-Chiesa.

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Lelli-Chiesa, M. Green’s Conjecture for curves on rational surfaces with an anticanonical pencil. Math. Z. 275, 899–910 (2013). https://doi.org/10.1007/s00209-013-1164-7

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  • DOI: https://doi.org/10.1007/s00209-013-1164-7

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