The birational geometry of Rg,2 and Prym-canonical divisorial strata

We prove that the moduli space of double covers ramified at two points Rg,2 R g , 2 is uniruled for 3≤ g≤ 6 3 ≤ g ≤ 6 and of general type for g≥ 16 g ≥ 16 . Furthermore, we consider Prym-canonical divisorial strata in the moduli space CⁿRg C n R ¯ g parametrizing n -pointed Prym curves, and we compute their classes in PicQ(CⁿRg) Pic Q ( C n R ¯ g ) .