The Stacks project

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 [vistoli_hilbert]

    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 [olsson-starr]

    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 [olsson_homstacks]

    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 ([alexeev-knutson])

    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 [lieblich_remarks]

    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 [starr_artin]

    As an application of the main result, a common generalization of Vistoli's Hilbert stack [vistoli_hilbert] and Alexeev and Knutson's stack of branchvarieties [alexeev-knutson] 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 [lundkvist-skjelnes]

    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.

Comments (1)

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.

There are also:

  • 2 comment(s) on Section 110.5: Papers in the literature

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