SpringerLink
Log in

Syzygies of Curves Beyond Green’s Conjecture

  • Conference paper
  • First Online:
Geometry of Moduli (Abelsymposium 2017)

Part of the book series: Abel Symposia ((ABEL,volume 14))

Included in the following conference series:

Abstract

We survey three results on syzygies of curves beyond Green’s conjecture, with a particular emphasis on drawing connections between the study of syzygies and other topics in moduli theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 160.49
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
EUR 213.99
Price includes VAT (Germany)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

  2. M. Aprodu, G. Farkas, Green’s conjecture for curves on arbitrary K3 surfaces. Comput. Math. 147(03), 839–851 (2011)

    MathSciNet  MATH  Google Scholar 

  3. M. Aprodu, J. Nagel, Koszul Cohomology and Algebraic Geometry, vol. 52 (American Mathematical Society, Providence, RI, 2010)

    MATH  Google Scholar 

  4. E. Arbarello, A. Bruno, Rank two vector bundles on polarised Halphen surfaces and the Gauss-Wahl map for du Val curves (2016). ar**v:1609.09256

    Google Scholar 

  5. E. Arbarello, A. Bruno, E. Sernesi, On hyperplane sections of K3 surfaces. Algebraic Geometry. ar**v:1507.05002 (to appear)

    Google Scholar 

  6. M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, M. Moeller, Compactification of strata of abelian differentials (2016). ar**v:1604.08834

    Google Scholar 

  7. A. Beauville, Prym varieties and the Schottky problem. Invent. Math. 41(2), 149–196 (1977)

    Article  MathSciNet  Google Scholar 

  8. A. Beauville, J.-Y. Mérindol, Sections hyperplanes des surfaces K3. Duke Math. J. 55(4), 873–878 (1987)

    Article  MathSciNet  Google Scholar 

  9. E. Bertini, Introduzione alla geometria proiettiva degli iperspazi: con appendice sulle curve algebriche e loro singolarità. Enrico Spoerri, Pisa (1907)

    Google Scholar 

  10. D. Buchsbaum, D. Eisenbud, What makes a complex exact? J. Algebra 25(2), 259–268 (1973)

    Article  MathSciNet  Google Scholar 

  11. D. Buchsbaum, D. Eisenbud, Generic free resolutions and a family of generically perfect ideals. Adv. Math. 18(3), 245–301 (1975)

    Article  MathSciNet  Google Scholar 

  12. A. Chiodo, Stable twisted curves and their r-spin structures. Ann. Inst. Fourier 58(5), 1635–1689 (2008)

    Article  MathSciNet  Google Scholar 

  13. A. Chiodo, G. Farkas, Singularities of the moduli space of level curves. J. Eur. Math. Soc. 19(3), 603–658 (2017)

    Article  MathSciNet  Google Scholar 

  14. A. Chiodo, D. Eisenbud, G. Farkas, F.-O. Schreyer, Syzygies of torsion bundles and the geometry of the level l modular variety over \(\overline {\mathcal {M}_g}\). Invent. Math. 194(1), 73–118 (2013)

    Google Scholar 

  15. E. Colombo, G. Farkas, A. Verra, C. Voisin, Syzygies of Prym and paracanonical curves of genus 8. Épijournal de Géométrie Algébrique 1(1) (2017). ar**v:1612.01026v2

    Google Scholar 

  16. O. Debarre, Sur le probleme de Torelli pour les variétés de Prym. Am. J. Math. 111(1), 111–134 (1989)

    Article  Google Scholar 

  17. I.V. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces. J. Math. Sci. 81(3), 2599–2630 (1996)

    Article  MathSciNet  Google Scholar 

  18. J. Eagon, D. Northcott, Ideals defined by matrices and a certain complex associated with them, in Proceedings of the Royal Society of London, vol. 269, pp. 188–204 (1962)

    Article  MathSciNet  Google Scholar 

  19. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150 (Springer, Berlin, 1995)

    Chapter  Google Scholar 

  20. D. Eisenbud, J. Harris, On varieties of minimal degree (A Centennial Account), in Proceedings of Symposium in Pure Mathematics, vol. 46, pp. 3–13 (1985)

    MathSciNet  MATH  Google Scholar 

  21. D: Eisenbud, H. Lange, G. Martens, F.-O. Schreyer, The Clifford dimension of a projective curve. Comput. Math. 72(2), 173–204 (1989)

    Google Scholar 

  22. D. Epema, Surfaces with canonical hyperplane sections, CWI tract 1. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam (1984)

    Google Scholar 

  23. G. Farkas, Syzygies of curves and the effective cone of \(\overline {M}_g\). Duke Math. J. 135(1), 53–98 (2006)

    Article  MathSciNet  Google Scholar 

  24. G. Farkas, Koszul divisors on moduli spaces of curves. Am. J. Math. 131(3), 819–867 (2009)

    Article  MathSciNet  Google Scholar 

  25. G. Farkas, M. Kemeny, (in preparation)

    Google Scholar 

  26. G. Farkas, M. Kemeny, Linear syzygies of curves with prescribed gonality (2016). ar**v:1610.04424

    Google Scholar 

  27. G. Farkas, M. Kemeny, The generic Green–Lazarsfeld secant conjecture. Invent. Math. 203(1), 265–301 (2016)

    Article  MathSciNet  Google Scholar 

  28. G. Farkas, M. Kemeny, The Prym-Green Conjecture for torsion bundles of high order. Duke Math. J. 166(6), 1103–1124 (2017)

    Article  MathSciNet  Google Scholar 

  29. G. Farkas, M. Kemeny, The resolution of paracanonical curves of odd genus (2017). ar**v:1707.06297

    Google Scholar 

  30. G. Farkas, K. Ludwig, The Kodaira dimension of the moduli space of Prym varieties. J. Eur. Math. Soc. 12(3), 755–795 (2010)

    Article  MathSciNet  Google Scholar 

  31. G. Farkas, R. Pandharipande, The moduli space of twisted canonical divisors. J. Inst. Math. Jussieu 1–58 (2016). https://doi.org/10.1017/S1474748016000128

    Article  MathSciNet  Google Scholar 

  32. G. Farkas, M. Popa, Effective divisors on \(\overline {\mathcal M}_g\), curves on K3 surfaces, and the Slope Conjecture. J. Algebraic Geom. 14(2), 241–267 (2005)

    Article  MathSciNet  Google Scholar 

  33. G. Farkas, N. Tarasca, Du Val curves and the pointed Brill–Noether theorem. Sel. Math. 23(3), 2243–2259 (2017)

    Article  MathSciNet  Google Scholar 

  34. G. Farkas, M. Mustaţă, M. Popa, Divisors on \(\mathcal {M}_{g, g+ 1}\) and the minimal resolution conjecture for points on canonical curves. Annales Scientifiques de l’École Normale Supérieure 36(4), 553–581 (2003)

    Article  MathSciNet  Google Scholar 

  35. M. Green, Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19(1), 125–171 (1984)

    Article  MathSciNet  Google Scholar 

  36. M. Green, R. Lazarsfeld, On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83(1), 73–90 (1986)

    Article  MathSciNet  Google Scholar 

  37. M. Green, R. Lazarsfeld, Special divisors on curves on a K3 surface. Invent. Math. 89, 357–370 (1987)

    Article  MathSciNet  Google Scholar 

  38. J. Harris, D. Mumford, On the Kodaira dimension of the moduli space of curves. Invent. Math. 67(1), 23–86 (1982)

    Article  MathSciNet  Google Scholar 

  39. A. Hirschowitz, S. Ramanan, New evidence for Green’s conjecture on syzygies of canonical curves. Annales Scientifiques de l’École Normale Supérieure 31(2), 145–152 (1998)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  41. D. Mumford, Prym varieties I. Contributions to analysis (a collection of papers dedicated to Lipman Bers) 325, 350 (1974)

    Google Scholar 

  42. F. Schreyer, Syzygies of canonical curves and special linear series. Math. Ann. 275(1), 105–137 (1986)

    Article  MathSciNet  Google Scholar 

  43. F.-O. Schreyer, Green’s conjecture for general p-gonal curves of large genus, in Algebraic Curves and Projective Geometry (Springer, Berlin, 1989), pp. 254–260

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  45. C. Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus. Comput. Math. 141(5), 1163–1190 (2005)

    MathSciNet  MATH  Google Scholar 

  46. J. Wahl, The Jacobian algebra of a graded Gorenstein singularity. Duke Math. J. 55(4), 843–511 (1987)

    Article  MathSciNet  Google Scholar 

  47. J. Wahl, Gaussian maps on algebraic curves. J. Differ. Geom. 32(1), 77–98 (1990)

    Article  MathSciNet  Google Scholar 

  48. J. Wahl, On the cohomology of the square of an ideal sheaf. J. Algebraic Geom. 76, 481–871 (1997)

    MathSciNet  MATH  Google Scholar 

  49. E. Witten, Algebraic geometry associated with matrix models of two-dimensional gravity, Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), pp. 235–269 (1993)

    Google Scholar 

Download references

Acknowledgements

It is a pleasure to thank the organisers of the Abel Symposium 2017 for a wonderful conference in a spectacular location. The results in this survey are joint work with my coauthor Gavril Farkas, who has taught me much of what I know about syzygies. I also thank D. Eisenbud and F.-O. Schreyer for enlightening conversations on these topics.

This survey is an amalgamation of material taken from my course on syzygies in Spring 2017 as well as talks given at UCLA and Berkeley in Autumn 2017. In particular, I thank Aaron Landesman for several corrections and improvements to my course notes. I also would like to thank the referee for the careful reading.

Author information

Authors and Affiliations

  1. Stanford University, Stanford, CA, USA

    Michael Kemeny

Authors
  1. Michael Kemeny
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Oslo, Oslo, Norway

    Jan Arthur Christophersen

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Kristian Ranestad

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kemeny, M. (2018). Syzygies of Curves Beyond Green’s Conjecture. In: Christophersen, J., Ranestad, K. (eds) Geometry of Moduli. Abelsymposium 2017. Abel Symposia, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-94881-2_7

Download citation

Publish with us

Policies and ethics

Search

Navigation