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Resolution of indeterminacy of rational maps to proper tame stacks

Myeong Jae Jeon

Abstract

We show the resolution of indeterminacy of rational maps from a regular surface to a tame stack locally of finite type over an excellent scheme. The proof uses the valuative criterion for proper tame morphisms, which was proved by Bresciani and Vistoli, together with the resolution of singularities for excellent surfaces and the root stack construction. Using Hironaka's results on the resolution of singularities over fields of characteristic zero, we extend the result to rational maps from a regular scheme of arbitrary dimension to a tame stack locally of finite type over a field of characteristic zero. We also provide a Purity Lemma for higher dimensional tame stacks, generalizing results of Abramovich, Olsson, and Vistoli, which also plays an essential role in the proof.

Resolution of indeterminacy of rational maps to proper tame stacks

Abstract

We show the resolution of indeterminacy of rational maps from a regular surface to a tame stack locally of finite type over an excellent scheme. The proof uses the valuative criterion for proper tame morphisms, which was proved by Bresciani and Vistoli, together with the resolution of singularities for excellent surfaces and the root stack construction. Using Hironaka's results on the resolution of singularities over fields of characteristic zero, we extend the result to rational maps from a regular scheme of arbitrary dimension to a tame stack locally of finite type over a field of characteristic zero. We also provide a Purity Lemma for higher dimensional tame stacks, generalizing results of Abramovich, Olsson, and Vistoli, which also plays an essential role in the proof.

Paper Structure

This paper contains 14 sections, 17 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Notations and Conventions
  3. Root Stacks
  4. The Purity Lemma
  5. Proofs of Theorems \ref{['thm:main']}, \ref{['thm:main-threefold']}, and \ref{['thm:main-higher-dim']}
  6. Proof of Theorem \ref{['thm:main']} (Reductions)
  7. Proof of Theorem \ref{['thm:main']} (Extension to codimension 2 components)
  8. Proof of Theorem \ref{['thm:main']} (Extension to codimension 1 components)
  9. Sketch of the proof of Theorem \ref{['thm:main-threefold']}
  10. Sketch of the proof of Theorem \ref{['thm:main-higher-dim']}
  11. Applications
  12. Morphisms at the generic point of regular varieties
  13. Extension of torsors
  14. Extension of fibrations

Key Result

Theorem 1.1

Let $f:\mathcal{M} \to \mathcal{N}$ be a tame, proper morphism of algebraic stacks locally of finite type over an excellent scheme $S$, with $\mathcal{M}$ separated over $S$. Let $X$ be a regular separated scheme of dimension $2$, locally of finite type over $S$, and $U \subseteq X$ a dense open sub Then, there exists a regular algebraic stack $\mathcal{X}$ of dimension $2$, locally of finite type

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Remark 2.1: Regularity can be checked flat locally
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 29 more