Projectivity of good moduli spaces of vector bundles on stacky curves

TL;DR

This paper proves the projectivity of moduli spaces of semistable vector bundles on stacky curves by applying good moduli space criteria and establishing ampleness of a determinantal line bundle, with applications to parabolic bundles.

Contribution

It demonstrates the existence and projectivity of moduli spaces of semistable vector bundles on stacky curves using new methods and provides effective bounds for line bundle properties.

Findings

The moduli stack of semistable vector bundles has a proper good moduli space.

A natural determinantal line bundle on this space is ample.

Effective bounds are provided for basepoint-freeness of line bundle powers.

Abstract

Moduli of vector bundles on stacky curves behave similarly to moduli of vector bundles on curves, except there are additional numerical invariants giving many different notions of stability. We apply the existence criterion for good moduli spaces of stacks to show that the moduli stack of semistable vector bundles on a stacky curve has a proper good moduli space. We moduli-theoretically prove that a natural determinantal line bundle on this moduli space is ample, thus proving this moduli space is projective. Our methods give effective bounds for when a power of this line bundle is basepoint-free. As a special case, we obtain new and effective constructions of moduli spaces of parabolic bundles.