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On base point free theorem and Mori dream spaces for log canonical threefolds over the algebraic closure of a finite field

Yusuke Nakamura, Jakub Witaszek

Abstract

The authors and D. Martinelli proved the base point free theorem for big line bundles on a three-dimensional log canonical projective pair defined over the algebraic closure of a finite field. In this paper, we drop the bigness condition when the characteristic is larger than five. Additionally, we discuss Mori dream spaces defined over the algebraic closure of a finite field.

On base point free theorem and Mori dream spaces for log canonical threefolds over the algebraic closure of a finite field

Abstract

The authors and D. Martinelli proved the base point free theorem for big line bundles on a three-dimensional log canonical projective pair defined over the algebraic closure of a finite field. In this paper, we drop the bigness condition when the characteristic is larger than five. Additionally, we discuss Mori dream spaces defined over the algebraic closure of a finite field.

Paper Structure

This paper contains 6 sections, 14 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main theorem
  4. Mori dream spaces over $\mathbb{\overline{F}}_p$
  5. Two-dimensional case
  6. Three-dimensional case

Key Result

Theorem \oldthetheorem

Let $(X, \Delta)$ be a three-dimensional projective log canonical pair defined over $\mathbb{\overline{F}}_p$, and let $D$ be a nef and big $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $X$. If $D - (K_X + \Delta)$ is also nef and big, then $D$ is semiample.

Theorems & Definitions (26)

  • Theorem \oldthetheorem: MNW
  • Theorem \oldthetheorem: Main theorem
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem: Artin, cf. Keel
  • Theorem \oldthetheorem: MNW
  • Lemma \oldthetheorem: cf. Keel
  • Proposition \oldthetheorem: BW
  • Theorem \oldthetheorem: BW
  • Lemma \oldthetheorem
  • ...and 16 more