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Birational involutions of the real projective plane fixing an irrational curve

Frédéric Mangolte

TL;DR

The paper analyzes birational involutions of the real and complex projective plane that fix an irrational curve, using a $G$-equivariant Mori framework with $G=\mathbb{Z}/2$. It shows how every order-$2$ birational map can be regularized to a biregular involution on a smooth rational surface and reduced to the $G$-equivariant classification of minimal $G$-surfaces, yielding a complete complex classification (linear, Geiser, Bertini, and de Jonquières) and a refined real landscape with 12 classes and intricate fixed-curve phenomena. A central invariant, the fixed curve $F(\tau)$, governs the complex classification, while real cases reveal richer behavior where fixed curves do not parametrize conjugacy classes. The authors also develop explicit birational models for $G$-conic bundles over $\mathbb{R}$, distinguishing de Jonquières and $d$-twisted Iskovskikh involutions and providing criteria for $G$-birational equivalence via discriminants and real loci, with corollaries describing non-conjugate families arising from hyperelliptic fixed curves.

Abstract

This review is an elaboration of a presentation given at the Real algebraic geometry and singularities conference in honor of Wojciech Kucharz's 70th birthday in Krakow in 2022.

Birational involutions of the real projective plane fixing an irrational curve

TL;DR

The paper analyzes birational involutions of the real and complex projective plane that fix an irrational curve, using a -equivariant Mori framework with . It shows how every order- birational map can be regularized to a biregular involution on a smooth rational surface and reduced to the -equivariant classification of minimal -surfaces, yielding a complete complex classification (linear, Geiser, Bertini, and de Jonquières) and a refined real landscape with 12 classes and intricate fixed-curve phenomena. A central invariant, the fixed curve , governs the complex classification, while real cases reveal richer behavior where fixed curves do not parametrize conjugacy classes. The authors also develop explicit birational models for -conic bundles over , distinguishing de Jonquières and -twisted Iskovskikh involutions and providing criteria for -birational equivalence via discriminants and real loci, with corollaries describing non-conjugate families arising from hyperelliptic fixed curves.

Abstract

This review is an elaboration of a presentation given at the Real algebraic geometry and singularities conference in honor of Wojciech Kucharz's 70th birthday in Krakow in 2022.

Paper Structure

This paper contains 9 sections, 9 theorems, 17 equations, 1 figure.

Table of Contents

  1. Examples of birational involutions of the projective plane $\mathbb{P}^2$
  2. Reduction to $G$-birational classification of minimal del Pezzo surfaces and conic bundles
  3. Main invariant: the fixed curve
  4. Main result
  5. Classification over $\mathbb{C}$
  6. Classification over $\mathbb{R}$
  7. Birational models of $G$-conic bundles over $\mathbb{R}$
  8. De Jonquières involutions
  9. $d$-twisted Iskovskikh involutions

Key Result

Proposition 1

Let $\Bbbk=\mathbb{R},\mathbb{C}$ and $\alpha\in\operatorname{Bir}_\Bbbk(\mathbb{P}^2)$ be an element of order $2$. There exists a smooth rational surface $S$ and a birational map defined over $\Bbbk$ such that $\tau:=\varphi^{-1}\alpha\varphi\in\operatorname{Aut}_\Bbbk(S)$ is a biregular involution of $S$.

Figures (1)

  • Figure 1: Singular irreducible real fiber

Theorems & Definitions (16)

  • Example 1: Standard Cremona involution
  • Example 2: Geiser involution
  • Proposition
  • Definition
  • Proposition
  • Proposition
  • Proposition
  • proof
  • Definition
  • Remark
  • ...and 6 more