Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1157–1164 | Cite as

The canonical syzygy conjecture for ribbons

  • Anand Deopurkar


Green’s canonical syzygy conjecture asserts a simple relationship between the Clifford index of a smooth projective curve and the shape of the minimal free resolution of its homogeneous ideal in the canonical embedding. We prove the analogue of this conjecture formulated by Bayer and Eisenbud for a class of non-reduced curves called ribbons. Our proof uses the results of Voisin and Hirschowitz–Ramanan on Green’s conjecture for general smooth curves.


Canonical syzygy conjecture Ribbons Green’s conjecture Koszul cohomology 

Mathematics Subject Classification

14H51 13D02 


  1. 1.
    Aprodu, M.: Remarks on syzygies of \(d\)-gonal curves. Math. Res. Lett. 12(2–3), 387–400 (2005). doi: 10.4310/MRL.2005.v12.n3.a9 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aprodu, M., Farkas, G.: Green’s conjecture for curves on arbitrary \(K3\) surfaces. Compos. Math. 147(3), 839–851 (2011). doi: 10.1112/S0010437X10005099 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aprodu, M., Farkas, G.: Green’s conjecture for general covers. In: Alexeev, V., Gibney, A., Izadi, E., Kollár, J., Looijenga, E. (eds.) Compact moduli spaces and vector bundles, contemporary mathematics, vol. 564, pp. 211–226. American Mathematical Society, Providence (2012). doi: 10.1090/conm/564/11147
  4. 4.
    Aprodu, M., Pacienza, G.: The Green conjecture for exceptional curves on a \(K3\) surface. Int. Math. Res. Not. IMRN 14(rnn043), 25 (2008). doi: 10.1093/imrn/rnn043 MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bayer, D., Eisenbud, D.: Ribbons and their canonical embeddings. Trans. AMS 347(3), 719–756 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deopurkar, A., Fedorchuk, M., Swinarski, D.: Toward GIT stability of syzygies of canonical curves. Algebr. Geom. 3(1), 1–22 (2016). doi: 10.14231/AG-2016-001 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ein, L., Lazarsfeld, R.: Asymptotic syzygies of algebraic varieties. Invent. Math. 190, 603–646 (2012). doi: 10.1007/s00222-012-0384-5 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eisenbud, D., Green, M.: Clifford indices of ribbons. Trans. Amer. Math. Soc. 347(3), 757–765 (1995). doi: 10.2307/2154872 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fong, L.Y.: Rational ribbons and deformation of hyperelliptic curves. J. Algebraic Geom. 2(2), 295–307 (1993)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Green, M., Lazarsfeld, R.: On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83(1), 73–90 (1986). doi: 10.1007/BF01388754 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Green, M.L.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19(1), 125–171 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hall, J.: Moduli of singular curves. arXiv:1011.6007 [math.AG] (2010)
  13. 13.
    Hartshorne, R.: Deformation Theory, Graduate Texts in Mathematics, vol. 257. Springer, New York (2010). doi: 10.1007/978-1-4419-1596-2 Google Scholar
  14. 14.
    Hirschowitz, A., Ramanan, S.: New evidence for Green’s conjecture on syzygies of canonical curves. Ann. Sci. École Norm. Sup. (4) 31(2), 145–152 (1998). doi: 10.1016/S0012-9593(98)80013-X MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lelli-Chiesa, M.: Green’s conjecture for curves on rational surfaces with an anticanonical pencil. Math. Z. 275(3–4), 899–910 (2013). doi: 10.1007/s00209-013-1164-7 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Schreyer, F.O.: Green’s conjecture for general \(p\)-gonal curves of large genus. In: Algebraic Curves and Projective Geometry (Trento, 1988), Lecture Notes in Math., vol. 1389, pp. 254–260. Springer, Berlin (1989). doi: 10.1007/BFb0085937
  17. 17.
    Teixidor, I., Bigas, M.: Green’s conjecture for the generic \(r\)-gonal curve of genus \(g\ge 3r-7\). Duke Math. J. 111(2), 195–222 (2000). doi: 10.1215/S0012-7094-02-11121-1 CrossRefzbMATHGoogle Scholar
  18. 18.
    Voisin, C.: Green’s generic syzygy conjecture for curves of even genus lying on a \(K3\) surface. J. Eur. Math. Soc. (JEMS) 4(4), 363–404 (2002). doi: 10.1007/s100970200042 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Voisin, C.: Green’s canonical syzygy conjecture for generic curves of odd genus. Compos. Math. 141(5), 1163–1190 (2005). doi: 10.1112/S0010437X05001387 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe University of GeorgiaAthensUSA

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