The degree of the Prym map for cyclic coverings

Given a finite morphism between smooth projective curves one can associate to it a Prym variety (an abelian variety, not necessarily principal). The corresponding Prym map is the map between the moduli space of coverings and the moduli space of abelian varieties with some fixed polarization type. By dimension reasons, only in very few cases one can expect the Prym map to be generically finite over its image.

In this talk I will recall some notorious examples where the Prym map is finite and the geometrical interpretation of the fiber. I also explain the case of étale cyclic coverings of degree 7 over a genus 2-curve. We show that the Prym map is generically finite over a special subvariety of the moduli space of 6-dimensional abelian varieties with polarization type (1,1,1,1,1,7). By extending the map to a proper map on a partial compactification of the space of coverings and performing a local analysis we compute that the degree of this Prym map is 10.

This a joint work with Herbert Lange.