Artin's axioms, composition and moduli spaces

TL;DR

This paper proves that Artin's axioms are compatible with composition of morphisms, establishing the algebraicity of certain stacks including generalizations of Hilbert stacks and branchvarieties.

Contribution

It demonstrates the compatibility of Artin's axioms with composition, leading to new algebraic stack constructions in moduli theory.

Findings

Artin's axioms are compatible with composition of 1-morphisms.

Some natural stacks are algebraic, including generalizations of Hilbert stacks.

The stack of branchvarieties is shown to be algebraic.

Abstract

We prove Artin's axioms satisfy a compatibility for composition of 1-morphisms of stacks in groupoids. Consequently, some natural stacks in groupoids are algebraic, including a common generalization of Vistoli's Hilbert stack and the stack of branchvarieties of Alexeev and Knutson.