The birational geometry of {R}₆, ₂ and Prym-canonical divisorial strata

Abstract We prove that the moduli space of double covers ramified at two points {R}₆, 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 {Cⁿ {R}}g C n R ¯ g parametrizing n -pointed Prym curves, and we compute their classes in Pic ₐ ({Cⁿ {R}}g) Pic Q (C n R ¯ g).