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Mori Fibrations in Mixed Characteristic

Liam Stigant

Abstract

This paper resolves several outstanding questions regarding the Minimal Model Program for klt threefolds in mixed characteristic. Namely termination for pairs which are not pseudo-effective, finiteness of minimal models and the Sarkisov Program.

Mori Fibrations in Mixed Characteristic

Abstract

This paper resolves several outstanding questions regarding the Minimal Model Program for klt threefolds in mixed characteristic. Namely termination for pairs which are not pseudo-effective, finiteness of minimal models and the Sarkisov Program.

Paper Structure

This paper contains 8 sections, 40 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Termination
  4. Relatively Log Terminal Pairs
  5. RLT Polytopes
  6. Finiteness of Log Terminal Models
  7. Geography of Ample Models
  8. Sarkisov Program

Key Result

Theorem 1.1

Let $X$ be an integral, normal threefold over $R$ equipped with a projective morphism $X \to T$, where $T$ is quasi-projective over $R$. If $(X,\Delta)$ is a threefold dlt pair over $R$ and the image of $X$ in $T$ is positive dimensional then any $K_{X}+\Delta$ MMP terminates.

Theorems & Definitions (82)

  • Theorem 1.1: \ref{['termination']}
  • Theorem 1.2: \ref{['sarkisov']}
  • Theorem 1.3: \ref{['rltfiniteness']}
  • Definition 1
  • Definition 2
  • Theorem 2.1: Cone Theorem
  • proof
  • Theorem 2.2: Basepoint Free Theorem
  • Theorem 2.3
  • Theorem 2.4
  • ...and 72 more