Finite generation of the algebra of type A conformal blocks via birational geometry

TL;DR

This paper proves that the algebra of type A conformal blocks is finitely generated by analyzing the birational geometry of moduli spaces of parabolic bundles, showing they are Mori dream spaces.

Contribution

It demonstrates the finite generation of the algebra of type A conformal blocks using Mori's program and describes the birational models via moduli spaces of parabolic bundles.

Findings

Moduli space of parabolic bundles is a Mori dream space.

Finite generation of the algebra of type A conformal blocks is established.

Effective cone and birational models are explicitly described.

Abstract

We study birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori's program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone which was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.