Koszul modules and Green’s conjecture


We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green’s conjecture for every g-cuspidal rational curve over an algebraically closed field \({{\mathbf {k}}}\), with \({\text {char}}({{\mathbf {k}}})=0\) or \({\text {char}}({{\mathbf {k}}})\ge \frac{g+2}{2}\). As a consequence, we deduce that the general canonical curve of genus g satisfies Green’s conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for \({{\mathfrak {s}}}{{\mathfrak {l}}}_2\)-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve.

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


  1. 1.

    Akin, K., Buchsbaum, D., Weyman, J.: Schur functors and Schur complexes. Adv. Math. 44, 207–278 (1982)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Aprodu, M., Farkas, G., Papadima, Ş., Raicu, C., Weyman, J.: Topological Invariants of Groups and Koszul Modules. arXiv:1806.01702

  3. 3.

    Aprodu, M., Nagel, J.: Koszul Cohomology and Algebraic Geometry. AMS University Lecture Series, vol. 52. American Mathematical Society (2010)

  4. 4.

    Bopp, C., Schreyer, F.-O.: A Version of Green’s Conjecture in Positive Characteristics. arXiv:1803.10481

  5. 5.

    Coskun, I., Riedl, E.: Normal Bundles of Rational Curves on Complete Intersections. arXiv:1705.08441

  6. 6.

    Dimca, A., Papadima, S., Suciu, A.: Topology and geometry of cohomology jump loci. Duke Math. J. 148, 405–457 (2009)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Ein, L., Lazarsfeld, R.: Tangent developable surfaces and syzygies of generic canonical curves, in preparation, to appear in the Bulletin of the AMS

  8. 8.

    Eisenbud, D.: Green’s conjecture: an orientation for algebraists. In: Free Resolutions in Commutative Algebra and Algebraic Geometry (Sundance 1990), Research Notes in Mathematics, vol. 2, pp. 51–78. Jones and Bartlett, Boston (1992)

  9. 9.

    Eisenbud, D.: The Geometry of Syzygies. A Second Course in Commutative Algebra and Algebraic Geometry. Graduate Texts in Mathematics, vol. 229. Springer, New York (2005)

    Google Scholar 

  10. 10.

    Eisenbud, D., Harris, J.: Limit linear series: basic theory. Invent. Math. 85, 337–371 (1986)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Eisenbud, D., Schreyer, F.-O.: Equations and Syzygies of K3 Carpets and Unions of Scrolls. arXiv:1804.08011

  12. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

    Fulton, W., Harris, J.: Representation theory. A first course. Graduate Texts in Mathematics, vol. 129. Springer, New York (1991)

    Google Scholar 

  14. 14.

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

    Article  Google Scholar 

  15. 15.

    Green, M., Lazarsfeld, R.: Higher obstructions to deforming cohomology groups of line bundles. J. Am. Math. Soc. 4, 87–103 (1991)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hartshorne, R.: Algebraic Geometry, Springer Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)

    Google Scholar 

  17. 17.

    Hartshorne, R.: Generalized divisors on Gorenstein curves and a theorem of Noether. J. Math. Kyoto Univ. 26, 375–386 (1986)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hassett, B., Hyeon, D.: Log canonical models for the moduli space of curves: the first divisorial contraction. Trans. Am. Math. Soc. 361, 4471–4489 (2009)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Hermite, C.: Sur la theorie des fonctions homogenes à deux indéterminées. Camb. Dublin Math. J. 9, 172–217 (1854)

    Google Scholar 

  20. 20.

    Jantzen, J.C.: Representations of Algebraic Groups, Pure and Applied Mathematics. Academic Press, New York (1987)

    Google Scholar 

  21. 21.

    Kaji, H.: On the tangentially degenerate curves. J. Lond. Math. Soc. 33, 430–440 (1986)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. Oxford University Press (1995)

  23. 23.

    Papadima, S., Suciu, A.: Vanishing resonance and representations of Lie algebras. J. Reine Angew. Math. 706, 83–101 (2015)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Schreyer, F.-O.: Syzygies of Curves with Special Pencils, Ph.D. Thesis, Brandeis University (1983)

  25. 25.

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

    MathSciNet  Article  Google Scholar 

  26. 26.

    Schubert, D.: A new compactification of the moduli space of curves. Compos. Math. 78, 297–313 (1991)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Smyth, D.: Modular compactifications of the space of pointed elliptic curves I. Compos. Math. 147, 877–913 (2011)

    MathSciNet  Article  Google Scholar 

  28. 28.

    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)

    MathSciNet  Article  Google Scholar 

  29. 29.

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

    MathSciNet  Article  Google Scholar 

  30. 30.

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

    MathSciNet  Article  Google Scholar 

  31. 31.

    Weyman, J.: Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Mathematics. Cambridge University Press (2003)

Download references


We acknowledge with thanks the contribution of A. Suciu. This project, including the companion paper [2], started with the paper [23], and since then we benefited from numerous discussions with him. We warmly thank A. Beauville, L. Ein, D. Eisenbud, B. Klingler, P. Pirola, F.-O. Schreyer and C. Voisin for interesting discussions related to this circle of ideas. We are particularly grateful to R. Lazarsfeld who read an early version of the paper and suggested many improvements which significantly clarified the exposition. Aprodu was partially supported by the Romanian Ministry of Research and Innovation, CNCS - UEFISCDI, grant PN-III-P4-ID-PCE-2016-0030, within PNCDI III. Farkas was supported by the DFG grant Syzygien und Moduli. Raicu was supported by the Alfred P. Sloan Foundation and by the NSF Grant No. 1600765. Weyman was partially supported by the Sidney Professorial Fund and the NSF grant No. 1802067.

Author information



Corresponding author

Correspondence to Gavril Farkas.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aprodu, M., Farkas, G., Papadima, Ş. et al. Koszul modules and Green’s conjecture. Invent. math. 218, 657–720 (2019). https://doi.org/10.1007/s00222-019-00894-1

Download citation