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Boundedness of weak Fano threefolds with fixed Gorenstein index in positive characteristic

Kenta Sato

Abstract

In this paper, we give a partial affirmative answer to the BAB conjecture for $3$-folds in characteristic $p>5$. Specifically, we prove that a set $\mathcal{D}$ of weak Fano $3$-folds over an uncountable algebraically closed field is bounded, if each element $X \in \mathcal{D}$ satisfies certain conditions regarding the Gorenstein index, a complement and Kodaira type vanishing. In the course of the proof, we also study a uniform lower bound for Seshadri constants of nef and big invertible sheaves on projective $3$-folds.

Boundedness of weak Fano threefolds with fixed Gorenstein index in positive characteristic

Abstract

In this paper, we give a partial affirmative answer to the BAB conjecture for -folds in characteristic . Specifically, we prove that a set of weak Fano -folds over an uncountable algebraically closed field is bounded, if each element satisfies certain conditions regarding the Gorenstein index, a complement and Kodaira type vanishing. In the course of the proof, we also study a uniform lower bound for Seshadri constants of nef and big invertible sheaves on projective -folds.
Paper Structure (27 sections, 69 theorems, 255 equations)

This paper contains 27 sections, 69 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and conventions
  4. Singularities
  5. Augmented base locus
  6. Seshadri constant
  7. Non-fixed effective divisors
  8. Spreading out
  9. Bounded family
  10. Non-vanishing and base point free theorems
  11. Models of pairs
  12. Lower bounds for Seshadri constants
  13. Two dimensional case
  14. Three dimensional case
  15. Birationally boundedness
  16. ...and 12 more sections

Key Result

Theorem A

Fix a DCC subset $I \subseteq [0,1] \cap \mathbb{Q}$, a rational number $\varepsilon>0$, integers $r,A>0$ and an uncountable algebraically closed field $k$ of characteristic larger than $5$. Suppose that $\mathcal{D} =\{(X_j, \Delta_j)\}_{j \in J}$ is a set of $3$-dimensional projective log pairs ov Then $\mathcal{D}$ is bounded.

Theorems & Definitions (152)

  • Conjecture 1.1: BAB conjecture
  • Theorem A: Theorem \ref{['bdd of weak Fano']}
  • Remark 1.2
  • Theorem B: Corollary \ref{['bb dim3']}
  • Theorem C: Theorem \ref{['Lower bound for seshadri']}
  • Theorem D: Theorem \ref{['criterion of lbb']}
  • Theorem E: Theorem \ref{['Bir bdd to bdd']}
  • Definition 2.1
  • Lemma 2.2: BMPS+
  • Remark 2.3: KM
  • ...and 142 more