Factorizations of Moduli Morphisms and Universal Maps to Deligne-Mumford Stacks

TL;DR

This paper investigates factorizations of moduli space morphisms of algebraic stacks, establishing universal maps to well-behaved stacks and analyzing conditions for stabilization of chains of such morphisms.

Contribution

It introduces a universal morphism to stacks with desirable properties and studies the stabilization of chains of adequate moduli space morphisms.

Findings

Existence of universal morphisms to stacks with modular properties.

Chains of adequate moduli space morphisms stabilize under certain conditions.

Stabilization can fail for general adequate moduli space morphisms.

Abstract

Let $\mathcal{X}$ be an algebraic stack admitting a moduli space $\mathcal{X}_{\mathrm{mod}}$. We study the factorizations of the moduli space morphism $\mathcal{X}\rightarrow\mathcal{X}_{\mathrm{mod}}$ to construct intermediate stacks that simplify the stacky structure of $\mathcal{X}$ while retaining more structural information than $\mathcal{X}_{\mathrm{mod}}$. Under mild assumptions, we prove the existence of a universal morphism from $\mathcal{X}$ to stacks satisfying well-behaved `modular properties' (such as being Deligne-Mumford, having finite inertia, or being uniformizable), and show that this universal map is itself an adequate moduli space morphism. We achieve this by proving that ascending chains of adequate moduli space morphisms from a Noetherian stack stabilize if they are cohomologically affine or with target Deligne-Mumford stacks. Finally, we demonstrate that stabilization completely fails for general adequate moduli space morphisms. We construct a simple Noetherian, Deligne-Mumford stack admitting an infinite, non-stabilizing chain of adequate moduli space morphisms, whose limit is a non-algebraic fpqc stack.