Counter-examples of high Clifford index to Prym-Torelli

Izadi, E.; Lange, H.

doi:10.1090/S1056-3911-2012-00587-6

Authors: E. Izadi and H. Lange
Journal: J. Algebraic Geom. 21 (2012), 769-787
DOI: https://doi.org/10.1090/S1056-3911-2012-00587-6
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

Arnaud Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149–196. MR 572974, DOI https://doi.org/10.1007/BF01418373
Arnaud Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 3, 309–391 (French). MR 472843
Arnaud Beauville, Sous-variétés spéciales des variétés de Prym, Compositio Math. 45 (1982), no. 3, 357–383 (French). MR 656611
Christina Birkenhake and Herbert Lange, Complex abelian varieties, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004. MR 2062673

G. Castelnuovo. Sulle serie algebriche di gruppi di punti appartenenti ad una curva algebrica. Rend. R. Acc. dei Lincei, 15(5):337–344, 1906.

Olivier Debarre, Sur le probleme de Torelli pour les varieties de Prym, Amer. J. Math. 111 (1989), no. 1, 111–134 (French). MR 980302, DOI https://doi.org/10.2307/2374482

Ron Donagi, The tetragonal construction, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 181–185. MR 598683, DOI https://doi.org/10.1090/S0273-0979-1981-14875-8

Ron Donagi, The fibers of the Prym map, Curves, Jacobians, and abelian varieties (Amherst, MA, 1990) Contemp. Math., vol. 136, Amer. Math. Soc., Providence, RI, 1992, pp. 55–125. MR 1188194, DOI https://doi.org/10.1090/conm/136/1188194

David Eisenbud, Herbert Lange, Gerriet Martens, and Frank-Olaf Schreyer, The Clifford dimension of a projective curve, Compositio Math. 72 (1989), no. 2, 173–204. MR 1030141

Robert Friedman and Roy Smith, The generic Torelli theorem for the Prym map, Invent. Math. 67 (1982), no. 3, 473–490. MR 664116, DOI https://doi.org/10.1007/BF01398932

T. Graber, J. Harris, and J. Starr. A note on Hurwitz schemes of covers of a positive genus curve. Arxiv:math/0205056.

E. Izadi, H. Lange, and V. Strehl, Correspondences with split polynomial equations, J. Reine Angew. Math. 627 (2009), 183–212. MR 2494932, DOI https://doi.org/10.1515/CRELLE.2009.015

V. I. Kanev, A global Torelli theorem for Prym varieties at a general point, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 2, 244–268, 431 (Russian). MR 651647

H. Lange and E. Sernesi, Quadrics containing a Prym-canonical curve, J. Algebraic Geom. 5 (1996), no. 2, 387–399. MR 1374713

I. G. Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962), 319–343. MR 151460, DOI https://doi.org/10.1016/0040-9383%2862%2990019-8

David Mumford, Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325–350. MR 0379510

Juan-Carlos Naranjo, Prym varieties of bi-elliptic curves, J. Reine Angew. Math. 424 (1992), 47–106. MR 1144161, DOI https://doi.org/10.1515/crll.1992.424.47

V. V. Shokurov, Algebraic curves and their Jacobians [ MR1060328 (91i:14022)], Algebraic geometry, III, Encyclopaedia Math. Sci., vol. 36, Springer, Berlin, 1998, pp. 219–270. MR 1602379, DOI https://doi.org/10.1007/978-3-662-03662-4_2

Alessandro Verra, The Prym map has degree two on plane sextics, The Fano Conference, Univ. Torino, Turin, 2004, pp. 735–759. MR 2112601

Gerald E. Welters, Recovering the curve data from a general Prym variety, Amer. J. Math. 109 (1987), no. 1, 165–182. MR 878204, DOI https://doi.org/10.2307/2374557

Additional Information

E. Izadi
Affiliation: Department of Mathematics, Boyd Graduate Studies Research Center, University of Georgia, Athens, Georgia 30602-7403
MR Author ID: 306461
Email: izadi@math.uga.edu

H. Lange
Affiliation: Department Mathematik, Universität Erlangen-Nürnberg, Bismarckstrasse 1 1/2, D-91054 Erlangen, Germany
Email: lange@mi.uni-erlangen.de

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.