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On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$

Xintong Jiang

TL;DR

This work advances the study of Fano-type varieties in positive characteristic by proving that boundedness is preserved under normalization, providing a practical canonical bundle formula for Fano-type fibrations in dimension up to three at large characteristic, and establishing effective birationality for certain strongly F-regular weak Fano varieties. It develops adjunction techniques, including divisorial adjunction and F-adjunction, to control discriminant and moduli parts and to construct complements, thereby extending birational tools into positive characteristic. Collectively, these results contribute partial progress toward the BAB conjecture in positive characteristic and offer a framework for obtaining uniform birational mappings via F-singularities and F-pure centers. The methods have potential implications for bounding families, constructing complements, and understanding the behavior of Fano-type fibrations under base change and contraction in characteristic $p>0$.

Abstract

In this paper, we give some results on the birational geometry of varieties of Fano type and boundedness problems in positive characteristic, including a result ensuring that boundedness is invariant under normalizations, a canonical bundle formula for fibrations of Fano type which is easier to use, and the effective birationality of certain weak Fano varieties with good singularities, which is predicted by the BAB conjecture.

On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$

TL;DR

This work advances the study of Fano-type varieties in positive characteristic by proving that boundedness is preserved under normalization, providing a practical canonical bundle formula for Fano-type fibrations in dimension up to three at large characteristic, and establishing effective birationality for certain strongly F-regular weak Fano varieties. It develops adjunction techniques, including divisorial adjunction and F-adjunction, to control discriminant and moduli parts and to construct complements, thereby extending birational tools into positive characteristic. Collectively, these results contribute partial progress toward the BAB conjecture in positive characteristic and offer a framework for obtaining uniform birational mappings via F-singularities and F-pure centers. The methods have potential implications for bounding families, constructing complements, and understanding the behavior of Fano-type fibrations under base change and contraction in characteristic .

Abstract

In this paper, we give some results on the birational geometry of varieties of Fano type and boundedness problems in positive characteristic, including a result ensuring that boundedness is invariant under normalizations, a canonical bundle formula for fibrations of Fano type which is easier to use, and the effective birationality of certain weak Fano varieties with good singularities, which is predicted by the BAB conjecture.
Paper Structure (14 sections, 21 theorems, 11 equations)

This paper contains 14 sections, 21 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Resolution of singularities
  4. Divisors, pairs and linear systems
  5. B-divisors and generalized pairs
  6. Singularities from MMP
  7. Contractions and minimal model programs
  8. $\mathbb Q$-factorialization
  9. Volumes, Kodaira dimensions and Iitaka fibrations
  10. Fano pairs and varieties of Fano type
  11. Bounded families
  12. F-singularities
  13. Adjunctions and F-adjunctions
  14. A weak version of effective birationality of Fano varieties

Key Result

Theorem 1.2

Let $k$ be an algebraically closed field, suppose that a family $\mathcal{P}$ of projective varieties $k$ is bounded, then the set $\mathcal{P}^\nu$ of normalization of the elements in $\mathcal{P}$ is also bounded.

Theorems & Definitions (42)

  • Conjecture 1.1: BAB Conjecture
  • Theorem 1.2: Theorem 2.4
  • Theorem 1.3: Canonical bundle formula for threefolds, Theorem 3.3
  • Corollary 1.4: Non-klt center adjunction, Theorem 3.5,3.6
  • Corollary 1.5: Corollary 3.7
  • Theorem 1.6: Effective birationality, Theorem 4.4
  • Corollary 1.7: Corollary 4.5
  • Theorem 2.1
  • proof
  • Theorem 2.2: c.f. birkar2014existenceflipsminimalmodelsbirkar2014existencemorifibrespaces
  • ...and 32 more