Prym varieties of cyclic coverings

Abstract

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.

On the dimension of totally geodesic submanifolds in the Prym loci

“…and it is surjective at the generic point if either r ≥ 3 and g ≥ 1, or r = 2 and g ≥ 3, r = 1 and g ≥ 5, r = 0 and g ≥ 6 (see [14]).…”

Rational points on abelian varieties over function fields and Prym varieties

“…For the classical case of Prym varieties of double coverings of curves, we refer to [1]. The following result gives an equivalent description of the Prym variety and shows furthermore that it coincides with the Prym variety of covers of curves (see [1,[6][7][8]) up to isogeny.…”

“…For curves, the Albanese variety coincides with the Jacobian variety. Hence if X and Y are curves, the Prym variety P (X/Y ) coincides with the Prym variety for covers of curves, see [6,7]. In this case, there are two fundamental homomorphisms: the norm homomorphism Nm f : Jac(X) → Jac(Y ) and the pull-back homomorphism f * : Jac(Y ) → Jac(X) and it holds that…”

On the Prym map for cyclic covers of genus two curves

“…Here we want to describe this differential, following [LO10]. Consider again a level curve [D, η] ∈ R g,d and let f : C → D be a corresponding cyclic cover.…”