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On moduli of foliated surfaces

Calum Spicer, Roberto Svaldi, Sebastian Velazquez

Abstract

We present a definition of stable family of foliations and show that the corresponding moduli functor for foliated surfaces is representable by a Deligne-Mumford stack.

On moduli of foliated surfaces

Abstract

We present a definition of stable family of foliations and show that the corresponding moduli functor for foliated surfaces is representable by a Deligne-Mumford stack.

Paper Structure

This paper contains 38 sections, 49 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Stable families of foliations
  3. Definition of the moduli functor
  4. Deformations of singularities
  5. Versality
  6. Boundedness (or lack thereof) of the moduli functor $\mathcal{M}^{2,1}$
  7. Infinitesimal structure of the moduli functor
  8. Comparison between moduli of stable foliations and of stable varieties.
  9. Foliation vs. integrable distribution
  10. Role of the Minimal Model Program
  11. Future directions
  12. Structure and sketch of proof
  13. Divisorial sheaves, Mumford divisors and relative Mumford divisors
  14. Integrable distributions
  15. Pull-back integrable distributions
  16. ...and 23 more sections

Key Result

Theorem 1.1

The moduli functor of stable foliated surfaces, denoted $\mathcal{M}^{2, 1}$ is represented by a Deligne-Mumford stack locally of finite type over $\mathbb C$ and which satisfies the valuative criterion for properness with respect to DVRs which are finite type over $\mathbb C$.

Theorems & Definitions (116)

  • Theorem 1.1: = Theorem \ref{['thm:rep2']}
  • Theorem 1.2: = Theorem \ref{['thm:rep']}
  • Theorem 1.3: = Theorem \ref{['thm_lc_deform']}
  • Theorem 1.4: = Theorem \ref{['thm_inversion_of_adjunction_full']}
  • Theorem 1.5
  • Theorem 1.6: = Theorem \ref{['thm_versalilty_stable_def']}
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 106 more