Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Universal flattening of Frobenius

Takehiko Yasuda

Abstract

For a variety $X$ of positive characteristic and a non-negative integer $e$, we define its $e$-th F-blowup to be the universal flattening of the $e$-iterated Frobenius of $X$. Thus we have the sequence (a set labeled by non-negative integers) of blowups of $X$. Under some condition, the sequence stabilizes and leads to a nice (for instance, minimal or crepant) resolution. For tame quotient singularities, the sequence leads to the $G$-Hilbert scheme.

Universal flattening of Frobenius

Abstract

For a variety of positive characteristic and a non-negative integer , we define its -th F-blowup to be the universal flattening of the -iterated Frobenius of . Thus we have the sequence (a set labeled by non-negative integers) of blowups of . Under some condition, the sequence stabilizes and leads to a nice (for instance, minimal or crepant) resolution. For tame quotient singularities, the sequence leads to the -Hilbert scheme.

Paper Structure

This paper contains 25 sections, 44 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Basic properties
  3. Construction
  4. Kunz's criterion for smoothness
  5. Compatibility
  6. Étale morphism
  7. Completion
  8. Product
  9. Smooth morphism
  10. Field extension
  11. Separation of analytic branches
  12. The resolution of singularities of a curve
  13. F-blowups of toric varieties
  14. Toric and Gröbner computation of the F-blowup
  15. Frobenius-like morphisms of toric varieites
  16. ...and 10 more sections

Key Result

Proposition 1.2

Theorems & Definitions (92)

  • Definition 1.1
  • Proposition 1.2
  • Remark 1.5
  • Theorem 1.6: Theorem \ref{['thm-curve-resolution']}
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9: Theorem \ref{['thm-GtoF-isom']}
  • Lemma 2.1
  • proof
  • Definition 2.2
  • ...and 82 more