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On monotonicity of F-blowup sequences

Takehiko Yasuda

Abstract

For each variety in positive characteristic, there is a series of canonically defined blowups, called F-blowups. We are interested in the question of whether the $e+1$-th blowup dominates the $e$-th, locally or globally. It is shown that the answer is affirmative (globally for any $e$) when the given variety is F-pure. As a corollary, we obtain some result on the stability of the sequence of F-blowups. We also give a sufficient condition for local domination.

On monotonicity of F-blowup sequences

Abstract

For each variety in positive characteristic, there is a series of canonically defined blowups, called F-blowups. We are interested in the question of whether the -th blowup dominates the -th, locally or globally. It is shown that the answer is affirmative (globally for any ) when the given variety is F-pure. As a corollary, we obtain some result on the stability of the sequence of F-blowups. We also give a sufficient condition for local domination.

Paper Structure

This paper contains 7 sections, 10 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm-F-pure']}
  4. On local domination by $\mathop{\mathrm{FB}}\nolimits_{e+1} X$ over $\mathop{\mathrm{FB}}\nolimits_{e} X$
  5. The toric case
  6. Stability of F-blowup sequences
  7. Proof of Proposition \ref{['prop-not-preserve']}

Key Result

Theorem 1.2

Suppose that $X$ is F-pure, that is, the natural morphism $\mathcal{O}_{X} \to (F_{1})_{*} \mathcal{O}_{X}$ locally splits as a morphism of $\mathcal{O}_{X}$-modules. Then the F-blowup sequence of $X$ is monotone.

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 2.1
  • Proposition 4.1
  • proof
  • proof : Proof of Theorem \ref{['thm-local-condition']}
  • Theorem 5.1
  • Definition 5.2
  • ...and 12 more