Birational involutions of the real projective plane

TL;DR

This paper classifies birational involutions of the real projective plane, revealing 12 classes over the reals and highlighting differences from the complex case.

Contribution

It provides a comprehensive classification of real birational involutions, including new examples and distinctions from complex classifications.

Findings

Identified 12 classes of involutions over the reals

Discovered fixed curves do not always determine conjugacy classes

Contrasts with the complex classification of involutions

Abstract

We classify birational involutions of the real projective plane up to conjugation. In contrast with an analogous classification over the complex numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, L. Bayle and A. Beauville), which includes 4 different classes of involutions, we discover 12 different classes over the reals, and provide many examples when the fixed curve of an involution does not determine its conjugacy class in the real plane Cremona group.