Prym varieties

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If C is an algebraic curve, and is a fixed point free involution on C, then the prym variety of the double cover is Connected component of Norm-1(0) containing the identity. (see p.297 of ref 1)

references and further reading

Arbarello, E.; Cornalba, M.; Griffiths, P. A.; Harris, J. Geometry of algebraic curves. Vol. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 267. Springer-Verlag, New York, 1985. xvi+386 pp. ISBN: 0-387-90997-4

Donagi, Ron; Smith, Roy Campbell The structure of the Prym map. Acta Math. 146 (1981), no. 1-2, 25--102.

Donagi, Ron The fibers of the Prym map. Curves, Jacobians, and abelian varieties (Amherst, MA, 1990), 55--125, Contemp. Math., 136, Amer. Math. Soc., Providence, RI, 1992.

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

arXiv:0804.4616 The Kodaira dimension of the moduli space of Prym varieties Gavril Farkas, Katharina Ludwig Journal of the European Mathematical Society