| Giving a finite
map between smooth coverings one can associate to it a polarized
abelian variety, its Prym variety. This procedure induces the Prym map
between the moduli space R_{p,g} of cyclic coverings of degree p over a
genus g curve and the moduli of abelian varieties with some fixed
polarization. There are exactly three cases when one can expect the
Prym map to be generically finite, namely (p,g)=(2,6),(3,4),(7,2). In
the case of double coverings of a genus 6 curve, R. Donagi proved that
the fibers have the structure of the 27 lines on smooth cubic surface.
C. Faber studied the case of triple cyclic coverings over a genus 4
curve and showed that Prym map is of degree 16. In this talk we will explain what is the corresponding picture for the remaining case of degree-7 cyclic coverings over a genus 2 curve. We will show that the Prym map is generically finite and can be extended to a proper map. We will also give a description of its image. This a joint work with Herbert Lange. |