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Birational complexity and conic fibrations

Joaquín Moraga

Abstract

Let $(X,B)$ be a log Calabi--Yau pair of dimension $n$, index one, and birational complexity $c$. We show that $(X,B)$ has a crepant birational model that admits a tower of Mori fiber spaces of which at least $n-c$ are conic fibrations. Motivated by the proof of the previous statement, we introduce new measures of the complexity of a log Calabi--Yau pair; the alteration complexity and the conic complexity. We characterize when these invariants are zero. Finally, we give applications of the tools of the main theorem to birational superrigidity, Fano hypersurfaces, dual complexes, Weil indices of Fano varieties, and klt singularities.

Birational complexity and conic fibrations

Abstract

Let be a log Calabi--Yau pair of dimension , index one, and birational complexity . We show that has a crepant birational model that admits a tower of Mori fiber spaces of which at least are conic fibrations. Motivated by the proof of the previous statement, we introduce new measures of the complexity of a log Calabi--Yau pair; the alteration complexity and the conic complexity. We characterize when these invariants are zero. Finally, we give applications of the tools of the main theorem to birational superrigidity, Fano hypersurfaces, dual complexes, Weil indices of Fano varieties, and klt singularities.
Paper Structure (27 sections, 48 theorems, 82 equations)

This paper contains 27 sections, 48 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Birational complexity and conic fibrations
  3. Alteration complexity
  4. Conic complexity
  5. Applications
  6. Preliminaries
  7. Equivariant generalized pairs
  8. Fano type varieties and log Calabi--Yau pairs
  9. Complexity measures for log Calabi--Yau pairs
  10. Negativity lemma
  11. Generalized canonical bundle formula
  12. The two ray game
  13. Finite equivariant geometry
  14. Finite actions on conic fibrations
  15. Generalized pairs on toric varieties
  16. ...and 12 more sections

Key Result

Theorem 1.1

(c.f. Theorem thm:cbir-conic) Let $(X,B)$ be a log Calabi--Yau pair of dimension $n$ and index one. Let $c:=c_{\rm bir}(X,B)$. Then $(X,B)$ has a crepant birational model that admits a tower of Mori fiber spaces of which at least $n-c$ are conic fibrations.

Theorems & Definitions (119)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 109 more