Projectivity of the moduli space of vector bundles on a curve

TL;DR

This paper revisits the classical proof of the projectivity of the moduli space of semistable vector bundles on a curve, proposing a modern approach using recent moduli space techniques and line bundle methods.

Contribution

It introduces a new, streamlined method combining good moduli space theory and determinantal line bundles to establish projectivity.

Findings

Modern approach successfully proves projectivity

Utilizes Faltings' argument with improvements

Provides a blueprint for future projectivity proofs

Abstract

We discuss the projectivity of the moduli space of semistable vector bundles on a curve of genus $g\geq 2$. This is a classical result from the 1960s, obtained using geometric invariant theory. We outline a modern approach that combines the recent machinery of good moduli spaces with determinantal line bundle techniques. The crucial step producing an ample line bundle follows an argument by Faltings with improvements by Esteves-Popa. We hope to promote this approach as a blueprint for other projectivity arguments.