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Existence of moduli spaces for algebraic stacks

Jarod Alper, Daniel Halpern-Leistner, Jochen Heinloth

Abstract

We provide necessary and sufficient conditions for when an algebraic stack admits a good moduli space and prove a semistable reduction theorem for points of algebraic stacks equipped with a $Θ$-stratification. These results provide a generalization of the Keel--Mori theorem to moduli problems whose objects have positive dimensional automorphism groups and give criteria on the moduli problem to have a separated or proper good moduli space. To illustrate our method, we apply these results to construct proper moduli spaces parameterizing semistable $\mathcal{G}$-bundles on curves and moduli spaces for objects in abelian categories.

Existence of moduli spaces for algebraic stacks

Abstract

We provide necessary and sufficient conditions for when an algebraic stack admits a good moduli space and prove a semistable reduction theorem for points of algebraic stacks equipped with a -stratification. These results provide a generalization of the Keel--Mori theorem to moduli problems whose objects have positive dimensional automorphism groups and give criteria on the moduli problem to have a separated or proper good moduli space. To illustrate our method, we apply these results to construct proper moduli spaces parameterizing semistable -bundles on curves and moduli spaces for objects in abelian categories.

Paper Structure

This paper contains 43 sections, 90 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reminder on local structure theorems for algebraic stacks
  4. Good and adequate moduli spaces
  5. Reminder on mapping stacks and filtrations
  6. The example of quotient stacks
  7. Valuative criteria for stacks
  8. Morphisms of stacks of filtrations
  9. Property Theta-P
  10. Theta-reductive morphisms
  11. Examples illustrating $\Theta$-reductivity
  12. Properties of $\Theta$-reductive morphisms
  13. Specialization of $k$-points
  14. Theta-surjective morphisms
  15. Elementary modifications and S-complete morphisms
  16. ...and 28 more sections

Key Result

Theorem A

Let $\cX$ be an algebraic stack of finite presentation, with affine stabilizers and separated diagonal over a noetherian algebraic space $S$ of characteristic $0$. Then $\cX$ admits a separated good moduli space $X$ if and only if $\cX$ is $\Theta$-reductive (D:theta-reductive) and S-complete (D:S-c

Theorems & Definitions (237)

  • Theorem A
  • Theorem B: \ref{['T:Langton-Algorithm']}
  • Theorem C: \ref{['C:valuative_criterion_semistable']}, \ref{['P:semistable_locus']}, \ref{['P:moduli_space_semistable_locus']}
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 227 more