The very good property for parabolic vector bundles over curves

TL;DR

This paper extends the concept of the "very good" property to moduli stacks of parabolic vector bundles on genus 0 and 1 curves, providing necessary and sufficient conditions for these cases.

Contribution

It generalizes Beilinson and Drinfeld's results by characterizing when the "very good" property holds for low-genus curves.

Findings

Necessary and sufficient conditions for the property in genus 0 and 1

Extension of the "very good" property to lower genus cases

Clarification of parabolic structures needed for the property

Abstract

The purpose of this note is to extend Beilinson and Drinfeld's "very good" property to moduli stacks of parabolic vector bundles on curves of genuses $g = 0$ and $g = 1$. Beilinson and Drinfeld show that for $g > 1$ a trivial parabolic structure is sufficient for the moduli stacks to be "very good". We give a necessary and sufficient condition on the parabolic structure for this property to hold in the case of lower genus.