It is known that, for a generic curve $C$, $End(J(C)) \cong \mathbb{Z}$, hence picard number is one. Is this true for Prym Variety? 



Picard number of Prym Variety
This is not true for a generic curve, but for a *very general* curve (= outside countably many strict subvarieties of the moduli space). The same is true for Prym varieties, because the closure of the locus of Prym varieties contains the Jacobian locus ("Wirtinger's construction").


