### 110.5.9 Hilbert, Quot, Hom and branchvariety stacks

Papers discussing Hilbert schemes and the like.

• Vistoli: The Hilbert stack and the theory of moduli of families

If $\mathcal{X}$ is a algebraic stack separated and locally of finite type over a locally noetherian and locally separated algebraic space $S$, Vistoli defines the Hilbert stack $\mathcal{H}\text{ilb}(\mathcal{F} / S)$ parameterizing finite and unramified morphisms from proper schemes. It is claimed without proof that $\mathcal{H}\text{ilb}(\mathcal{F} / S)$ is an algebraic stack. As a consequence, it is proved that with $\mathcal{X}$ as above, the Hom stack $\mathcal{H} \text{om}_ S(T, \mathcal{X})$ is an algebraic stack if $T$ is proper and flat over $S$.
• Olsson, Starr: Quot functors for Deligne-Mumford stacks

If $\mathcal{X}$ is a Deligne-Mumford stack separated and locally of finite presentation over an algebraic space $S$ and $\mathcal{F}$ is a locally finitely-presented $\mathcal{O}_\mathcal {X}$-module, the quot functor $\text{Quot}(\mathcal{F} / \mathcal{X} / S)$ is represented by an algebraic space separated and locally of finite presentation over $S$. This paper also defines generating sheaves and proves existence of a generating sheaf for tame, separated Deligne-Mumford stacks which are global quotient stacks of a scheme by a finite group.
• Olsson: Hom-stacks and Restrictions of Scalars

Suppose $\mathcal{X}$ and $\mathcal{Y}$ are Artin stacks locally of finite presentation over an algebraic space $S$ with finite diagonal with $\mathcal{X}$ proper and flat over $S$ such that fppf-locally on $S$, $\mathcal{X}$ admits a finite finitely presented flat cover by an algebraic space (eg. $\mathcal{X}$ is Deligne-Mumford or a tame Artin stack). Then $\mathop{\mathrm{Hom}}\nolimits _ S(\mathcal{X}, \mathcal{Y})$ is an Artin stack locally of finite presentation over $S$.
• Alexeev and Knutson: Complete moduli spaces of branchvarieties ()

They define a branchvariety of $\mathbf{P}^ n$ as a finite morphism $X \rightarrow \mathbf{P}^ n$ from a reduced scheme $X$. They prove that the moduli stack of branchvarieties with fixed Hilbert polynomial and total degrees of $i$-dimensional components is a proper Artin stack with finite stabilizer. They compare the stack of branchvarieties with the Hilbert scheme, Chow scheme and moduli space of stable maps.
• Lieblich: Remarks on the stack of coherent algebras

This paper constructs a generalization of Alexeev and Knutson's stack of branch-varieties over a scheme $Y$ by building the stack as a stack of algebras over the structure sheaf of $Y$. Existence proofs of $\text{Quot}$ and $\mathop{\mathrm{Hom}}\nolimits$ spaces are given.
• Starr: Artin's axioms, composition, and moduli spaces

As an application of the main result, a common generalization of Vistoli's Hilbert stack and Alexeev and Knutson's stack of branchvarieties is provided. If $\mathcal{X}$ is an algebraic stack locally of finite type over an excellent scheme $S$ with finite diagonal, then the stack $\mathcal{H}$ parameterizing morphisms $g: T \rightarrow \mathcal{X}$ from a proper algebraic space $T$ with a $G$-ample line bundle $L$ is an Artin stack locally of finite type over $S$.
• Lundkvist and Skjelnes: Non-effective deformations of Grothendieck's Hilbert functor

Shows that the Hilbert functor of a non-separated scheme is not represented since there are non-effective deformations.
• Halpern-Leistner and Preygel: Mapping stacks and categorical notions of properness [HL-P]

This paper gives a proof that the Hom stack is algebraic under some hypotheses on source and target which are more general than, or at least different from, the ones in Olsson's paper.

Comment #1595 by Matthew Emerton on

It might be good to add a reference to the paper "Mapping stacks and categorical notions of properness", by Halpern-Leistner and Preygel, here. It gives a proof that the Hom stack is algebraic under some hypotheses on source and target which are more general than, or at least different from, the ones in Olsson's paper.

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