Line bundles on the moduli stack of parahoric bundles

TL;DR

This paper studies line bundles on the moduli stack of parahoric bundles over a curve, establishing criteria for descent, proving a conjecture, and explicitly computing the Picard group in examples.

Contribution

It translates the problem into twisted conformal blocks, proves a conjecture of Pappas and Rapoport, and computes the Picard group explicitly in certain cases.

Findings

Criteria for line bundle descent established

Conjecture of Pappas and Rapoport proved

Explicit Picard group calculations in examples

Abstract

In this paper we investigate line bundles on $\mathrm{Bun}_{\mathcal{G}}$ the moduli stack of parahoric Bruhat--Tits bundles over a smooth projective curve. Translating this problem into one concerning twisted conformal blocks, we are able to establish criteria that detect when line bundles on an appropriate flag variety descend to $\mathrm{Bun}_{\mathcal{G}}$. Along the way we establish a conjecture of Pappas and Rapoport which describes sections of line bundles on $\mathrm{Bun}_{\mathcal{G}}$ using representation-theoretical means. We conclude the paper with examples where our methods allow us to explicitly determine the Picard group of $\mathrm{Bun}_{\mathcal{G}}$.