The Hilbert Stack

TL;DR

This paper introduces the Hilbert stack, an algebraic stack that parametrizes proper algebraic stacks with quasi-finite maps to a given algebraic stack, extending known results even for schemes.

Contribution

It establishes the existence of the Hilbert stack for algebraic stacks, generalizing classical Hilbert schemes to a broader stack context.

Findings

The Hilbert stack is algebraic under certain conditions.

It parametrizes proper algebraic stacks with quasi-finite maps.

This result was previously unknown even for schemes.

Abstract

Let \pi : X -> S be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers. We prove that there is an algebraic S-stack, the Hilbert stack, parameterizing proper algebraic stacks mapping quasi-finitely to X. This was previously unknown, even for a morphism of schemes.