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Singularities of Foliations and Good Moduli Spaces of Algebraic Stacks

Federico Bongiorno

Abstract

Drawing on the theory of Minimal Model Program singularities for foliations, we define relative canonical and log-canonical singularities for algebraic stacks with finite generic stabilisers. We show that if a point has log-canonical singularities, its stabiliser group is a finite extension of an algebraic torus, thus, étale locally, the good moduli space exists. If the singularity is canonical, we further show that the locus of stable points is non-empty.

Singularities of Foliations and Good Moduli Spaces of Algebraic Stacks

Abstract

Drawing on the theory of Minimal Model Program singularities for foliations, we define relative canonical and log-canonical singularities for algebraic stacks with finite generic stabilisers. We show that if a point has log-canonical singularities, its stabiliser group is a finite extension of an algebraic torus, thus, étale locally, the good moduli space exists. If the singularity is canonical, we further show that the locus of stable points is non-empty.

Paper Structure

This paper contains 23 sections, 34 theorems, 55 equations, 1 table.

Table of Contents

  1. Singularities of Foliations on Schemes
  2. Foliations on Schemes
  3. Foliations on Logarithmic Schemes
  4. Foliations on Discrete Valuation Rings
  5. Discrepancies of Foliations
  6. Singularities of Foliations
  7. Ascent and Descent
  8. Modifications and Alterations
  9. Normal Representations
  10. Semistable morphisms
  11. Singularities of Foliations on Stacks
  12. Foliations on Deligne--Mumford Stacks
  13. Relative Singularities of Algebraic Stacks
  14. Singularities and Base Change
  15. Resolution of Singularities
  16. ...and 8 more sections

Key Result

Theorem 1

Let $X$ be a Noetherian normal scheme over $\mathbb{C}$ and let $x \in X$ be a closed point. Suppose that $\partial$ is a local derivation of $X$ such that $x$ is $\partial$-invariant. Then $\partial$ is log-canonical at $x$ if and only if the induced endomorphism of the tangent space $T_{X, x}$ is

Theorems & Definitions (100)

  • Theorem : McQuillan--Panazzolo
  • Conjecture : Cascini--Spicer
  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Lemma 1.8
  • ...and 90 more