The line bundles on the stack of parabolic $G$-bundles over curves and their sections

TL;DR

This paper computes the Picard group of the moduli stack of quasi-parabolic G-bundles over curves, relates sections to conformal blocks, and explores properties of the canonical sheaf and coarse moduli spaces, extending known theorems.

Contribution

It explicitly calculates the Picard group, identifies sections with conformal blocks, and analyzes the canonical sheaf and factoriality of coarse moduli spaces for semi-stable bundles.

Findings

Picard group of the moduli stack is computed.

Sections correspond to conformal blocks of Tsuchiya, Ueno, and Yamada.

Coarse moduli spaces of semi-stable SO_r-bundles are not locally factorial for r ≥ 7.

Abstract

Let $X$ be a smooth, complete and connected curve and $G$ be a simple and simply connected algebraic group over $\comp$. We calculate the Picard group of the moduli stack of quasi-parabolic $G$-bundles and identify the spaces of sections of its members to the conformal blocs of Tsuchiya, Ueno and Yamada. We describe the canonical sheaf on these stacks and show that they admit a unique square root, which we will construct explicitly. Finally we show how the results on the stacks apply to the coarse moduli spaces and recover (and extend) the Drezet-Narasimhan theorem. We show moreover that the coarse moduli spaces of semi-stable $SO_r$-bundles are not locally factorial for $r\geq 7$.