Construction of the Moduli Space of Vector Bundles on an Orbifold Curve

TL;DR

This paper constructs the moduli space of semistable vector bundles on orbifold curves over any algebraically closed field, offering a GIT-free approach that generalizes known characteristic zero results.

Contribution

It introduces a new GIT-free method for constructing moduli spaces of vector bundles on orbifold curves over arbitrary characteristic fields.

Findings

Moduli space is a finite union of irreducible projective varieties.

Construction works in any characteristic, extending known characteristic zero results.

When non-empty, the moduli space has a well-understood geometric structure.

Abstract

Let $k$ be an algebraically closed field of any characteristic, and let $(X,P)$ be an orbifold curve over $k$. We construct the moduli space $\mathrm{M}_{(X,P)}^{\mathrm{ss}}(n, \Delta)$ of $P$-semistable bundles on $(X,P)$ of rank $n$ and determinant $\Delta$. In the characteristic zero case, this result is well known and follows from GIT techniques. Our construction follows a different approach inspired by a GIT-free construction of Faltings. We show that when the moduli space is non-empty, it is a finite disjoint union of irreducible projective varieties.