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Toroidal and toric models of fibrations over curves

Caucher Birkar

TL;DR

The paper develops a bounded toroidal/toric framework for fibrations over curves by blending de Jong’s nodal-curve theory with toric geometry, avoiding loss of relative boundedness that occurs in resolution-based toroidalisation. It introduces and exploits towers of couples, universal boundedness results, and special toric towers to translate toroidal problems into toric ones, while maintaining strict degree and boundary controls. The main contributions include a bounded toroidalisation theorem for fibrations over curves and a construction of bounded toric models, with detailed machinery to pull back toric towers through nodal-tower structures. These results enable reductions to toric settings in birational geometry and have potential applications to Fano fibrations and the boundedness of complements. The methods rely on alterations, nodal-curve techniques from de Jong, and a robust use of toric and toroidal geometry to preserve boundedness throughout the process.

Abstract

We construct relatively bounded toroidal and toric models of relatively bounded fibrations over curves.

Toroidal and toric models of fibrations over curves

TL;DR

The paper develops a bounded toroidal/toric framework for fibrations over curves by blending de Jong’s nodal-curve theory with toric geometry, avoiding loss of relative boundedness that occurs in resolution-based toroidalisation. It introduces and exploits towers of couples, universal boundedness results, and special toric towers to translate toroidal problems into toric ones, while maintaining strict degree and boundary controls. The main contributions include a bounded toroidalisation theorem for fibrations over curves and a construction of bounded toric models, with detailed machinery to pull back toric towers through nodal-tower structures. These results enable reductions to toric settings in birational geometry and have potential applications to Fano fibrations and the boundedness of complements. The methods rely on alterations, nodal-curve techniques from de Jong, and a robust use of toric and toroidal geometry to preserve boundedness throughout the process.

Abstract

We construct relatively bounded toroidal and toric models of relatively bounded fibrations over curves.

Paper Structure

This paper contains 28 sections, 21 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Morphisms
  4. Divisors, degree, and volume
  5. Pairs and singularities
  6. Fibre products
  7. Factoring morhisms
  8. Étale morphisms
  9. Couples and toroidal geometry
  10. Couples
  11. Universal families of relatively bounded families of couples
  12. Morphisms and towers of couples
  13. Toric geometry
  14. Formally Cartier divisors
  15. Toroidal couples
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $d,r\in \mathbb N$. Then there exists $r'\in \mathbb N$ depending only on $d,r$ satisfying the following. Assume that Then, perhaps after shrinking $Z$ around $z$, there exists a commutative diagram \xymatrix{ (X',D') \ar[d]^{f'} \ar[r]^{\pi} & (X,D) \ar[d]^f \\ (Z',E') \ar[r]^{\mu} & Z }of couples and a very ample$/Z'$ divisor $A'$ on $X'$ such that

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.5
  • proof
  • Lemma 2.7
  • proof
  • Lemma 2.9
  • proof
  • Lemma 2.10
  • proof
  • ...and 32 more