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The group of birational selfmaps of a conic fibration is uncountable

Enrica Floris

TL;DR

The paper proves that for any fibration $f:X\to Y$ with general fibre a rational curve, the group $\mathrm{Bir}(X/Y)$ is uncountable, and thus $\mathrm{Bir}(X)$ is uncountable when $X$ is birational to a conic bundle. It builds on and refines the BLZ approach by constructing uncountably many involutions distinguished by their indeterminacy loci, using isotrivial-fibration structure and a non-birationality criterion for complete intersections to yield a homomorphism $\mathrm{Bir}(X/Y)\to\bigoplus_I\mathbb{Z}/2$ with uncountable $I$. This demonstrates that birational selfmaps can form a richly structured, uncountable group in this geometric setting, contrasting with some known rigidity results in low dimensions.

Abstract

We prove that the group of birational selfmaps of a variety birational to a conic bundle is uncountable.

The group of birational selfmaps of a conic fibration is uncountable

TL;DR

The paper proves that for any fibration with general fibre a rational curve, the group is uncountable, and thus is uncountable when is birational to a conic bundle. It builds on and refines the BLZ approach by constructing uncountably many involutions distinguished by their indeterminacy loci, using isotrivial-fibration structure and a non-birationality criterion for complete intersections to yield a homomorphism with uncountable . This demonstrates that birational selfmaps can form a richly structured, uncountable group in this geometric setting, contrasting with some known rigidity results in low dimensions.

Abstract

We prove that the group of birational selfmaps of a variety birational to a conic bundle is uncountable.
Paper Structure (2 sections, 6 theorems, 7 equations)

This paper contains 2 sections, 6 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:Intro']}

Key Result

Theorem 1.1

Let $X, Y$ be projective varieties, let $f\colon X\to Y$ be a fibration whose general fibre is a rational curve. Then $\mathop{\mathrm{Bir}}\nolimits(X/Y)$ is uncountable.

Theorems & Definitions (11)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 1 more