Distinguishing Algebraic Spaces from Schemes

TL;DR

This paper develops local invariants to distinguish algebraic spaces from schemes and provides criteria to identify when moduli spaces are schemes, with applications to principal bundles on curves.

Contribution

It introduces new invariants for algebraic spaces and stacks, enabling topological criteria to determine when their moduli spaces are schemes.

Findings

Invariants effectively distinguish algebraic spaces from schemes.

Criteria successfully identify when moduli spaces are schemes.

Application to moduli of principal bundles on curves.

Abstract

We introduce local invariants of algebraic spaces and stacks which measure how far they are from being a scheme. Using these invariants, we develop mostly topological criteria to determine when the moduli space of a stack is a scheme. As an application we study moduli of principal bundles on a smooth projective curve.