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

Abstract

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.

Compactifications of moduli of $G$-bundles and conformal blocks

“…For G = SL(V ), this was proven by Belkale-Gibney in [5] and generalized to the case of parabolic bundles on pointed curves (using completely new methods) by Moon-Yoo in [21]. Our proof is based closely on Belkale-Gibney's, the main point being to show that A(C 0 ) is isomorphic (in sufficiently high degree) to the section ring of an ample line bundle on a normalized moduli space of singular G-bundles (theorem 5.4).…”

Positivity of the Poincaré bundle on the moduli space of vector bundles and its applications

“…Moreover, the birational geometry of M(r, L, m, a) is governed by the wall-crossing: Every rational contraction that appears in Mori's program can be described in terms of wall-crossings or their degenerations -the forgetful map (Example 2.7) and generalized Hecke correspondences (Remark 2.10). Consult [MY20,MY21] to see a more general framework.…”