Automorphism group of the moduli space of parabolic vector bundles with fixed degree

TL;DR

This paper characterizes all isomorphisms and 3-birational maps between moduli spaces of parabolic vector bundles with fixed degree, revealing their structure and a Torelli-type theorem linking the moduli space to the underlying curve.

Contribution

It provides a complete description of 3-birational maps between these moduli spaces and establishes a Torelli-type theorem relating the moduli space's class to the curve's isomorphism class.

Findings

All 3-birational maps are compositions of known transformations.

The 3-birational class of the moduli space determines the curve's isomorphism class.

Explicit description of automorphisms of the moduli space.

Abstract

We find all possible isomorphisms and 3-birational maps (i.e., birational maps which induce an isomorphism between open subsets whose respective complements have codimension at least 3) between moduli spaces of parabolic vector bundles with fixed degree. We prove that every 3-birational map can be described as a composition of tensorization by a fixed line bundle, Hecke transformations, dualization, taking pullback by an isomorphism between the curves and the action of the group of automorphisms of the Jacobian variety of the curve which fix the r-torsion. In particular, we prove a Torelli type theorem, stating that the 3-birational class of the moduli space determines the isomorphism class of the curve.