nLab Prym-Tyurin variety

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

$$\iota^\ast \Theta = n \Xi \,.$$

Here $n$ is called the exponent of $Z$ in $Jac(C)$.

(e.g. Birkenhage-Lange)

Properties

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

Related concepts

• Prym variety

References

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