Rank $2$ quasiparabolic vector bundles on $\mathbb{P}^1$ and the variety of linear subspaces contained in two odd-dimensional quadrics

TL;DR

This paper establishes an isomorphism between a moduli space of stable quasiparabolic vector bundles on the projective line and a variety of linear subspaces in an intersection of quadrics, linking vector bundle theory and algebraic geometry.

Contribution

It provides a new geometric description of the moduli space of quasiparabolic bundles using linear subspaces in quadrics, extending previous work on vector bundles on hyperelliptic curves.

Findings

N is isomorphic to the variety of (g-2)-dimensional linear subspaces in P^{2g}

The moduli space corresponds to linear subspaces contained in the intersection of two quadrics

The proof connects quasiparabolic bundles on P^1 with invariant bundles on hyperelliptic curves.

Abstract

Let N be the moduli space of stable rank 2 quasiparabolic vector bundles of fixed degree on the projective line with 2g+1 marked points, where g>1, and stability is with respect to the weights {0,1/2} at each marked point. In this note we show that N is isomorphic to the variety of (g-2)-dimensional linear subspaces of P^{2g}, contained in the intersection of two quadrics. The proof relies on the work of Bhosle on the relation among quasiparabolic vector bundles on P^1 and invariant vector bundles on hyperelliptic curves, and the description by Bhosle and Ramanan of the moduli space of stable rank 2 vector bundles on a hyperelliptic curve, with fixed determinant of odd degree.