The birational geometry of $$\overline{{\mathcal {R}}}_{g,2}$$ and Prym-canonical divisorial strata

Abstract

AbstractWe prove that the moduli space of double covers ramified at two points $${\mathcal {R}}_{g,2}$$ R g , 2 is uniruled for $$3\le g\le 6$$ 3 ≤ g ≤ 6 and of general type for $$g\ge 16$$ g ≥ 16 . Furthermore, we consider Prym-canonical divisorial strata in the moduli space $$\overline{{\mathcal {C}}^n{\mathcal {R}}}_g$$ C n R ¯ g parametrizing n-pointed Prym curves, and we compute their classes in $$\textrm{Pic}_{\mathbb {Q}}(\overline{{\mathcal {C}}^n{\mathcal {R}}}_g)$$ Pic Q ( C n R ¯ g ) .