Finite generation of the algebra of type A conformal blocks via birational geometry II: higher genus

We prove finite generation of the algebra of type A conformal blocks over arbitrary stable curves of any genus. As an application, we construct a flat family of irreducible normal projective varieties over the moduli stack of stable pointed curves, whose fiber over a smooth curve is a moduli space of semistable parabolic bundles. This generalizes a construction of a degeneration of the moduli space of vector bundles presented in a recent work of Belkale and Gibney.