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On the Resolutions of Non-Dicritical Foliations

Philip J. Carter

Abstract

We introduce the jet schemes of a holomorphic foliation, and thereby prove an alternate characterisation of simple singularities of codimension-$1$ foliations, independent of any normal form. This leads to an equivalent condition for the existence of a desingularisation in the non-dicritical case. We then prove that such a desingularisation always exists, at least on the level of germs.

On the Resolutions of Non-Dicritical Foliations

Abstract

We introduce the jet schemes of a holomorphic foliation, and thereby prove an alternate characterisation of simple singularities of codimension- foliations, independent of any normal form. This leads to an equivalent condition for the existence of a desingularisation in the non-dicritical case. We then prove that such a desingularisation always exists, at least on the level of germs.
Paper Structure (23 sections, 63 theorems, 47 equations)

This paper contains 23 sections, 63 theorems, 47 equations.

Table of Contents

  1. Conventions and Background
  2. Formal Schemes
  3. Prorings
  4. Formal Schemes via Prorings
  5. Other Results
  6. Jet Spaces
  7. Jets of Schemes
  8. Jets of Formal Schemes
  9. Foliations
  10. Foliations by Vector Fields
  11. Foliations by $1$-forms
  12. Pullback Foliations
  13. Jet Spaces of Foliations
  14. Basic Definitions
  15. Tangent Schemes
  16. ...and 8 more sections

Key Result

Proposition 2.4

The category of prorings has all projective limits.

Theorems & Definitions (195)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Proposition 2.4
  • proof : Proof:
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 185 more