Generic Torelli theorem for Prym varieties of ramified coverings

We consider the Prym map from the space of double coverings of a curve of genus g
with r branch points to the moduli space of abelian varieties. We prove that P : Rg;r :
Ag generically injective if
r > 6 and g > 2; r = 6 and g > 3; r = 4 and g > 5 or r = 2 and g > 6:
We also show that a very general Prym variety of dimension at least 4 is not isogenous. The research that lead to the present paper was partially supported by a grant of the group GNSAGA of INdAM
to a Jacobian.