Skip to main content
SpringerLink
Log in

The generic Green–Lazarsfeld Secant Conjecture

Inventiones mathematicae volume 203, pages 265–301 (2016)Cite this article

Abstract

Using lattice theory on special \(K3\) surfaces, calculations on moduli stacks of pointed curves and Voisin’s proof of Green’s Conjecture on syzygies of canonical curves, we prove the Prym–Green Conjecture on the naturality of the resolution of a general Prym-canonical curve of odd genus, as well as (many cases of) the Green–Lazarsfeld Secant Conjecture on syzygies of non-special line bundles on general curves.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. 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 

  2. Aprodu, M., Farkas, G.: Green’s Conjecture for general covers, compact moduli spaces and vector bundles. In: Alexeev, V., et al. (eds), Contemporary Mathematics, vol. 564, pp. 211–226 (2012)

  3. Arbarello, E., Cornalba, M.: The Picard group of the moduli space of curves. Topology 26, 153–171 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of Algebraic Curves, Grundlehren der mathematischen Wissenschaften, vol. I. Springer, New York (1985)

    Book  Google Scholar 

  5. Chiodo, A., Eisenbud, D., Farkas, G., Schreyer, F.-O.: Syzygies of torsion bundles and the geometry of the level \(\ell \) modular varieties over \(\overline{\cal M}_g\). Invent. Math. 194, 73–118 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dolgachev, I.: Mirror symmetry for lattice polarized \(K3\) surfaces. J. Math. Sci. 81, 2599–2630 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Farkas, G.: Koszul divisors on moduli spaces of curves. Am. J. Math. 131, 819–869 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Farkas, G.: Brill–Noether with ramification at unassigned points. J. Pure Appl. Algebra 217, 1838–1843 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Farkas, G., Mustaţă, M., Popa, M.: Divisors on \({\cal M}_{g, g+1}\) and the minimal resolution conjecture for points on canonical curves. Annales Sci. de L’École Normale Supérieure 36, 553–581 (2003)

    Article  MATH  Google Scholar 

  10. Farkas, G., Ludwig, K.: The Kodaira dimension of the moduli space of Prym varieties. J. Eur. Math. Soc. 12, 755–795 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Farkas, G., Verra, A.: Moduli of theta-characteristics via Nikulin surfaces. Math. Ann. 354, 465–496 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fulton, W., Harris, J., Lazarsfeld, R.: Excess linear series on an algebraic curve. Proc. Am. Math. Soc. 92, 320–322 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Garbagnati, A., Sarti, A.: Projective models of \(K3\) surfaces with an even set. Adv. Geom. 8, 413–440 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. van Geemen, B., Sarti, A.: Nikulin involutions on \(K3\) surfaces. Math. Z. 255, 731–753 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Green, M.: Koszul cohomology and the cohomology of projective varieties. J. Differ. Geom. 19, 125–171 (1984)

    MATH  Google Scholar 

  16. Green, M., Lazarsfeld, R.: On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83, 73–90 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Green, M., Lazarsfeld, R.: Some results on the syzygies of finite sets and algebraic curves. Compos. Math. 67, 301–314 (1988)

    MathSciNet  MATH  Google Scholar 

  18. Harris, J., Mumford, D.: On the Kodaira dimension of \(\overline{\cal M}_g\). Invent. Math. 67, 23–88 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huybrechts, D.: Lectures on \(K3\) surfaces. http://www.math.uni-bonn.de/people/huybrech/K3Global

  20. Kemeny, M.: The moduli of singular curves on K3 surfaces. J. de Mathematiqes Pures et Appliquees. arXiv:1401.1047 (to appear)

  21. Koh, J., Stillman, M.: Linear syzygies and line bundles on an algebraic curve. J. Algebra 125, 120–132 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. Lazarsfeld, R.: A Sampling of Vector Bundle Techniques in the Study of Linear Series, Lectures on Riemann Surfaces (Trieste, 1987), World Scientific, pp. 500–559

  24. Mayer, A.: Families of \(K3\) surfaces. Nagoya Math. J. 48, 1–17 (1972)

    MathSciNet  Google Scholar 

  25. Schreyer, F.-O.: Syzygies of canonical curves and special linear series. Math. Ann. 275, 105–137 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We are grateful to M. Aprodu, D. Eisenbud, J. Harris, R. Lazarsfeld, F.-O. Schreyer, and especially to C. Voisin for many useful discussions related to this circle of ideas.

Author information

Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Gavril Farkas & Michael Kemeny

Authors
  1. Gavril Farkas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Kemeny
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gavril Farkas.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Farkas, G., Kemeny, M. The generic Green–Lazarsfeld Secant Conjecture. Invent. math. 203, 265–301 (2016). https://doi.org/10.1007/s00222-015-0595-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-015-0595-7

search

Navigation