Abelsymposium 2017: Geometry of Moduli pp 195-216 | Cite as

Syzygies of Curves Beyond Green’s Conjecture

  • Michael Kemeny
Conference paper
Part of the Abel Symposia book series (ABEL, volume 14)


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.



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.


  1. 1.
    M. Aprodu, Remarks on Szyzgies of d-gonal curves. Math. Res. Lett. 12, 387–400 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Aprodu, G. Farkas, Green’s conjecture for curves on arbitrary K3 surfaces. Comput. Math. 147(03), 839–851 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    M. Aprodu, J. Nagel, Koszul Cohomology and Algebraic Geometry, vol. 52 (American Mathematical Society, Providence, RI, 2010)zbMATHGoogle Scholar
  4. 4.
    E. Arbarello, A. Bruno, Rank two vector bundles on polarised Halphen surfaces and the Gauss-Wahl map for du Val curves (2016). arXiv:1609.09256Google Scholar
  5. 5.
    E. Arbarello, A. Bruno, E. Sernesi, On hyperplane sections of K3 surfaces. Algebraic Geometry. arXiv:1507.05002 (to appear)Google Scholar
  6. 6.
    M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, M. Moeller, Compactification of strata of abelian differentials (2016). arXiv:1604.08834Google Scholar
  7. 7.
    A. Beauville, Prym varieties and the Schottky problem. Invent. Math. 41(2), 149–196 (1977)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Beauville, J.-Y. Mérindol, Sections hyperplanes des surfaces K3. Duke Math. J. 55(4), 873–878 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    E. Bertini, Introduzione alla geometria proiettiva degli iperspazi: con appendice sulle curve algebriche e loro singolarità. Enrico Spoerri, Pisa (1907)Google Scholar
  10. 10.
    D. Buchsbaum, D. Eisenbud, What makes a complex exact? J. Algebra 25(2), 259–268 (1973)MathSciNetCrossRefGoogle Scholar
  11. 11.
    D. Buchsbaum, D. Eisenbud, Generic free resolutions and a family of generically perfect ideals. Adv. Math. 18(3), 245–301 (1975)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Chiodo, Stable twisted curves and their r-spin structures. Ann. Inst. Fourier 58(5), 1635–1689 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. Chiodo, G. Farkas, Singularities of the moduli space of level curves. J. Eur. Math. Soc. 19(3), 603–658 (2017)MathSciNetCrossRefGoogle Scholar
  14. 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. 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). arXiv:1612.01026v2Google Scholar
  16. 16.
    O. Debarre, Sur le probleme de Torelli pour les variétés de Prym. Am. J. Math. 111(1), 111–134 (1989)CrossRefGoogle Scholar
  17. 17.
    I.V. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces. J. Math. Sci. 81(3), 2599–2630 (1996)MathSciNetCrossRefGoogle Scholar
  18. 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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150 (Springer, Berlin, 1995)CrossRefGoogle Scholar
  20. 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)MathSciNetzbMATHGoogle Scholar
  21. 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. 22.
    D. Epema, Surfaces with canonical hyperplane sections, CWI tract 1. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam (1984)Google Scholar
  23. 23.
    G. Farkas, Syzygies of curves and the effective cone of \(\overline {M}_g\). Duke Math. J. 135(1), 53–98 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    G. Farkas, Koszul divisors on moduli spaces of curves. Am. J. Math. 131(3), 819–867 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    G. Farkas, M. Kemeny, (in preparation)Google Scholar
  26. 26.
    G. Farkas, M. Kemeny, Linear syzygies of curves with prescribed gonality (2016). arXiv:1610.04424Google Scholar
  27. 27.
    G. Farkas, M. Kemeny, The generic Green–Lazarsfeld secant conjecture. Invent. Math. 203(1), 265–301 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    G. Farkas, M. Kemeny, The Prym-Green Conjecture for torsion bundles of high order. Duke Math. J. 166(6), 1103–1124 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    G. Farkas, M. Kemeny, The resolution of paracanonical curves of odd genus (2017). arXiv:1707.06297Google Scholar
  30. 30.
    G. Farkas, K. Ludwig, The Kodaira dimension of the moduli space of Prym varieties. J. Eur. Math. Soc. 12(3), 755–795 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    G. Farkas, R. Pandharipande, The moduli space of twisted canonical divisors. J. Inst. Math. Jussieu 1–58 (2016). MathSciNetCrossRefGoogle Scholar
  32. 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)MathSciNetCrossRefGoogle Scholar
  33. 33.
    G. Farkas, N. Tarasca, Du Val curves and the pointed Brill–Noether theorem. Sel. Math. 23(3), 2243–2259 (2017)MathSciNetCrossRefGoogle Scholar
  34. 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)MathSciNetCrossRefGoogle Scholar
  35. 35.
    M. Green, Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19(1), 125–171 (1984)MathSciNetCrossRefGoogle Scholar
  36. 36.
    M. Green, R. Lazarsfeld, On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83(1), 73–90 (1986)MathSciNetCrossRefGoogle Scholar
  37. 37.
    M. Green, R. Lazarsfeld, Special divisors on curves on a K3 surface. Invent. Math. 89, 357–370 (1987)MathSciNetCrossRefGoogle Scholar
  38. 38.
    J. Harris, D. Mumford, On the Kodaira dimension of the moduli space of curves. Invent. Math. 67(1), 23–86 (1982)MathSciNetCrossRefGoogle Scholar
  39. 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)MathSciNetCrossRefGoogle Scholar
  40. 40.
    R. Lazarsfeld, Brill–Noether–Petri without degenerations. J. Differ. Geom. 23, 299–307 (1986)MathSciNetCrossRefGoogle Scholar
  41. 41.
    D. Mumford, Prym varieties I. Contributions to analysis (a collection of papers dedicated to Lipman Bers) 325, 350 (1974)Google Scholar
  42. 42.
    F. Schreyer, Syzygies of canonical curves and special linear series. Math. Ann. 275(1), 105–137 (1986)MathSciNetCrossRefGoogle Scholar
  43. 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. 254–260CrossRefGoogle Scholar
  44. 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)MathSciNetCrossRefGoogle Scholar
  45. 45.
    C. Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus. Comput. Math. 141(5), 1163–1190 (2005)MathSciNetzbMATHGoogle Scholar
  46. 46.
    J. Wahl, The Jacobian algebra of a graded Gorenstein singularity. Duke Math. J. 55(4), 843–511 (1987)MathSciNetCrossRefGoogle Scholar
  47. 47.
    J. Wahl, Gaussian maps on algebraic curves. J. Differ. Geom. 32(1), 77–98 (1990)MathSciNetCrossRefGoogle Scholar
  48. 48.
    J. Wahl, On the cohomology of the square of an ideal sheaf. J. Algebraic Geom. 76, 481–871 (1997)MathSciNetzbMATHGoogle Scholar
  49. 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

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Michael Kemeny
    • 1
  1. 1.Stanford UniversityStanfordUSA

Personalised recommendations