Geometriae Dedicata

, Volume 150, Issue 1, pp 391–403 | Cite as

Prym varieties of cyclic coverings

Original Paper


The Prym map of type (g, n, r) associates to every cyclic covering of degree n of a curve of genus g ramified at a reduced divisor of degree r the corresponding Prym variety. We show that the corresponding map of moduli spaces is generically finite in most cases. From this we deduce the dimension of the image of the Prym map.


Prym variety Prym map 

Mathematics Subject Classification (2000)

14H40 14H30 


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  1. 1.
    Bardelli F., Ciliberto C., Verra A.: Curves of minimal genus on a general abelian variety. Compos. Mathem. 96, 115–147 (1995)MathSciNetMATHGoogle Scholar
  2. 2.
    Barth W., Peters C., Van de Ven A.: Compact Complex Surfaces. Ergebnisse der Math. 4. Springer, Berlin (1984)Google Scholar
  3. 3.
    Beauville A.: Variétés de Prym et Jacobiennes intermediares. Annales Ec. Norm. Sup. 3, 309–391 (1977)MathSciNetGoogle Scholar
  4. 4.
    Birkenhake, Ch., Lange, H.: Complex Abelian Varieties. Second edition, Grundlehren der Math. Wiss. 302. Springer (2004)Google Scholar
  5. 5.
    Butler D.C.: Global sections and tensor products of line bundles over a curve. Math. Z. 231, 397–407 (1999)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Donagi R.: The tetragonal construction. Bull. Am. Soc. 4, 181–185 (1981)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Friedman R., Smith R.: The generic Torelli theorem for the Prym map. Invent. Math. 67, 473–490 (1982)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Green M., Lazarsfeld R.: On the projectivity normality of complete linear series on an algebraic curve. Invent. Math. 83, 73–90 (1986)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Harris J., Morrison I.: Moduli of Curves. GTM, No. 187. Springer, New York (1998)Google Scholar
  10. 10.
    Kanev V.: The global Torelli theorem for Prym varieties at a generic point. Math. USSR-Izv. 20, 235–258 (1983)MATHCrossRefGoogle Scholar
  11. 11.
    Lange H., Sernesi E.: On the Hilbert scheme of a Prym variety. Ann. di Matem. 183, 375–386 (2004)MathSciNetMATHGoogle Scholar
  12. 12.
    Sernesi E.: Deformations of Algebraic Schemes. Grundlehren der Math Wiss. 302. Springer, New York (2006)Google Scholar
  13. 13.
    Tamagawa A.: Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental groups. J. Alg. Geom. 13, 675–724 (2004)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany
  2. 2.Fachbereich MathematikUniversität Duisburg-EssenEssenGermany

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