Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Counter-examples of high Clifford index to Prym-Torelli

Authors: E. Izadi and H. Lange
Journal: J. Algebraic Geom. 21 (2012), 769-787
Published electronically: March 5, 2012
MathSciNet review: 2957696
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Abstract | References | Additional Information

Abstract: We give examples of curves of arbitrarily high Clifford index such that the Prym map is not injective at any of their étale double covers.

References [Enhancements On Off] (What's this?)

  • 1. A. Beauville.
    Prym varieties and the Schottky problem.
    Inventiones Math., 41:149-196, 1977. MR 0572974 (58:27995)
  • 2. A. Beauville.
    Variétés de Prym et jacobiennes intermédiaires.
    Annales Sc. de l'École Norm. Sup., 4ème série, 10:309-391, 1977. MR 0472843 (57:12532)
  • 3. A. Beauville.
    Sous-variétés spéciales des variétés de Prym.
    Compositio Math., 45, Fasc. 3:357-383, 1982. MR 656611 (83f:14025)
  • 4. C. Birkenhake and H. Lange.
    Complex abelian varieties, volume 302 of
    Grundlehren der Mathematischen Wissenschaften.
    Springer, Berlin, 2004. MR 2062673 (2005c:14001)
  • 5. G. Castelnuovo.
    Sulle serie algebriche di gruppi di punti appartenenti ad una curva algebrica.
    Rend. R. Acc. dei Lincei, 15(5):337-344, 1906.
  • 6. O. Debarre.
    Sur le problème de Torelli pour les variétés de Prym.
    Amer. Journal of Math., 111(1):111-134, 1989. MR 980302 (90b:14035)
  • 7. R. Donagi.
    The tetragonal construction.
    Bull. of the A.M.S., 4(2):181-185, 1981. MR 598683 (82a:14009)
  • 8. R. Donagi.
    The fibers of the Prym map.
    In Curves, Jacobians, and Abelian Varieties (Amherst, MA, 1990), volume 136 of Contemp. Math., pages 55-125. Amer. Math. Soc., 1992. MR 1188194 (94e:14037)
  • 9. D. Eisenbud, H. Lange, G. Martens, and F.O. Schreyer.
    The Clifford dimension of a projective curve.
    Comp. Math., 72(2):173-204, 1989. MR 1030141 (91b:14033)
  • 10. R. Friedman and R. Smith.
    The generic Torelli theorem for the Prym map.
    Invent. Math., 67(3):473-490, 1982. MR 664116 (83i:14017)
  • 11. T. Graber, J. Harris, and J. Starr.
    A note on Hurwitz schemes of covers of a positive genus curve.
  • 12. E. Izadi, H. Lange, and V. Strehl.
    Correspondences with split polynomial equations.
    J. Reine Angew. Math., 627:183-212, 2009. MR 2494932 (2010g:14038)
  • 13. V. Kanev.
    A global Torelli theorem for Prym varieties at a general point.
    Izv. Akad. Nauk. SSSR Ser. Mat., 46(2):244-268, 431, 1982. MR 651647 (83h:14022)
  • 14. H. Lange and E. Sernesi.
    Quadrics containing a Prym-canonical curve.
    Journal of Alg. Geom., 5(2):387-399, 1996. MR 1374713 (96k:14021)
  • 15. I. G. Macdonald.
    Symmetric products of an algebraic curve.
    Topology, 1:319-343, 1962. MR 0151460 (27:1445)
  • 16. D. Mumford.
    Prym varieties I.
    In L.V. Ahlfors, I. Kra, B. Maskit, and L. Niremberg, editors, Contributions to Analysis, pages 325-350. Academic Press, 1974. MR 0379510 (52:415)
  • 17. J.-C. Naranjo.
    Prym varieties of bi-elliptic curves.
    J. Reine Angew. Math., 424:47-106, 1992. MR 1144161 (93b:14068)
  • 18. V.V. Shokurov.
    Algebraic curves and their Jacobians.
    In A.N. Parshin and I.R. Shafarevich, editors, Algebraic Geometry III, volume 36 of Encycl. Math. Sciences, pages 219-261. Springer, Berlin, 1987. MR 1602379
  • 19. A. Verra.
    The Prym map has degree $ 2$ on plane sextics.
    In The Fano Conference, pages 735-759. Univ. Torino, Turin, 2004. MR 2112601 (2005k:14057)
  • 20. G. E. Welters.
    Recovering the curve data from a general Prym variety.
    Amer. J Math., 109(1):165-182, 1987. MR 878204 (88c:14041)

Additional Information

E. Izadi
Affiliation: Department of Mathematics, Boyd Graduate Studies Research Center, University of Georgia, Athens, Georgia 30602-7403

H. Lange
Affiliation: Department Mathematik, Universität Erlangen-Nürnberg, Bismarckstrasse 1 1/2, D-91054 Erlangen, Germany

Received by editor(s): February 2, 2010
Received by editor(s) in revised form: February 9, 2011
Published electronically: March 5, 2012
Additional Notes: The first author was partially supported by the National Security Agency. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author and do not necessarily reflect the views of the National Security Agency
Part of this work was done while the second author was visiting the University of Georgia; he would like to thank the University of Georgia for its hospitality.

American Mathematical Society