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On rational chain connectedness of globally +-regular varieties

Emre Alp Özavcı, Zsolt Patakfalvi, Kevin Tucker, Joe Waldron, Zheng Xu

Abstract

We prove that globally $+$-regular varieties are rationally chain connected in dimension three and mixed characteristic with residue field characteristic $p>5$. We also introduce a notion of strongly globally $+$-regular, and show that varieties of arbitrary dimension which are strongly globally $+$-regular over a dense open subset of $\mathrm{Spec}(\mathbb{Z})$ are rationally chain connected.

On rational chain connectedness of globally +-regular varieties

Abstract

We prove that globally -regular varieties are rationally chain connected in dimension three and mixed characteristic with residue field characteristic . We also introduce a notion of strongly globally -regular, and show that varieties of arbitrary dimension which are strongly globally -regular over a dense open subset of are rationally chain connected.
Paper Structure (11 sections, 24 theorems, 22 equations)

This paper contains 11 sections, 24 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions
  4. Globally $+$-regular varieties
  5. Log minimal model program
  6. Rational chain connectedness
  7. Strongly globally $+$-regular
  8. Definitions and basic properties
  9. Examples
  10. Rationally chain connectedness of globally $+$-regular varieties in mixed characteristic
  11. Rational connectedness of globally $+$-regular varieties in positive characteristic

Key Result

Theorem 1.1

Let $(X,\Delta)$ be a projective globally $+$-regular pair surjective over $V=\mathop{\mathrm{Spec}}\nolimits(R)$ where $R$ is a DVR of mixed characteristic with residue characteristic $p>5$. Suppose $\dim(X)\leq 3$. Then $X$ is rationally chain connected over $V$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1
  • Lemma 1
  • Proposition 1
  • Lemma 2
  • proof
  • Lemma 3
  • Theorem 2.1
  • ...and 37 more