nLab
Prym-Tyurin variety

Contents

Idea

The concept of Prym-Tyurin variety or generalized Prym variety is a slight generalization of that of Jacobian variety of an algebraic curve.

It is an abelian variety ZZ equipped with principal polarization Ξ\Xi such that there exists an algebraic curve CC and an embedding ι:ZJac(C)\iota \colon Z \hookrightarrow Jac(C) of ZZ into its Jacobian variety such that the pullback of the principal polarization Θ\Theta of Jac(C)Jac(C) along this embedding is an integral fraction of Ξ\Xi:

ι *Θ=nΞ. \iota^\ast \Theta = n \Xi \,.

Here nn is called the exponent of ZZ in Jac(C)Jac(C).

(e.g. Birkenhage-Lange)

Properties

Any principally polarized abelian variety is a Prym-Tyurin variety for a curve CC of sufficiently high genus.

References

  • Christina Birkenhake Herbert Lange, section 12.2 of Complex abelian varieties

Created on October 19, 2014 at 16:33:38. See the history of this page for a list of all contributions to it.