Compactified moduli of projective bundles

TL;DR

This paper develops a new method for compactifying moduli stacks of projective bundles, demonstrating irreducibility and virtual classes for surfaces, and providing explicit boundary descriptions via a derived Skolem-Noether theorem.

Contribution

It introduces a novel approach to compactifying stacks of PGL_n-torsors and describes boundary points explicitly using derived category techniques.

Findings

Moduli spaces are shown to be irreducible.

Constructed natural virtual fundamental classes.

Provided explicit boundary descriptions in the compactification.

Abstract

We present a method for compactifying stacks of $\PGL_n$-torsors (Azumaya algebras) on algebraic spaces. In particular, when the ambient space is a smooth projective surface we use our methods to show that various moduli spaces are irreducible and carry natural virtual fundamental classes. We also prove a version of the Skolem-Noether theorem for certain algebra objects in the derived category, which allows us to give an explicit description of the boundary points in our compactified moduli problem.