[Paper Review] Birational involutions of P^2

Lionel Bayle, Arnaud Beauville|ArXiv.org|Jul 6, 1999

Algebraic Geometry and Number Theory4 references57 citations

TL;DR

This paper provides a complete and precise classification of birational involutions of the projective plane ℙ² up to conjugacy, showing that every such involution is conjugate to exactly one of three types: De Jonquières, Geiser, or Bertini involutions. The classification relies on Mori theory and the study of biregular involutions on rational surfaces, with fixed curves serving as key invariants that parametrize conjugacy classes.

ABSTRACT

We give a "modern" version, based on Mori theory, of the classification of birational involutions of P^2 up to conjugacy. The result has been known for more than one century but the classical proofs are not always convincing.

Motivation & Objective

• To provide a complete and non-redundant classification of birational involutions of ℙ² up to conjugacy.
• To resolve the limitations of prior classifications by excluding singular fixed curves and ensuring uniqueness.
• To establish a precise correspondence between conjugacy classes of involutions and isomorphism classes of algebraic curves via their normalized fixed curves.
• To use Mori theory and biregular geometry on rational surfaces to reduce the classification problem to minimal pairs (S,σ).

Proposed method

• Reduces the classification of birational involutions of ℙ² to classifying minimal pairs (S,σ), where S is a rational surface and σ is a biregular involution.
• Applies Mori theory to analyze the Picard group fixed by σ, distinguishing cases based on the rank of Pic(S)^σ.
• Identifies two main cases: when Pic(S)^σ has rank >1 (leading to De Jonquières involutions via base-point-free pencils), and when rank=1 (leading to Geiser and Bertini involutions via lattice-theoretic arguments).
• Uses elementary transformations and blow-ups/downs to normalize configurations, reducing to standard surfaces like ℙ², ℱ₁, or Del Pezzo surfaces.
• Constructs explicit birational models by resolving singularities and analyzing fixed loci, particularly focusing on curves with ordinary multiple points.
• Establishes a one-to-one correspondence between conjugacy classes and isomorphism classes of curves via normalized fixed curves.

Experimental results

Research questions

• RQ1 What are the complete and non-redundant conjugacy classes of birational involutions of ℙ²?
• RQ2 How can the classification be made precise by excluding singular fixed curves that cause redundancy in prior work?
• RQ3 What role do fixed curves play in parametrizing conjugacy classes of birational involutions?
• RQ4 Which geometric structures (e.g., pencils, singularities, canonical models) characterize the three types of involutions?
• RQ5 How do the Geiser and Bertini involutions arise from double covers of Del Pezzo surfaces and singular quadrics?

Key findings

• Every birational involution of ℙ² is conjugate to exactly one of three types: De Jonquières, Geiser, or Bertini involutions.
• De Jonquières involutions of degree d correspond bijectively to isomorphism classes of hyperelliptic curves of genus d−2 for d≥3.
• Geiser involutions correspond bijectively to isomorphism classes of non-hyperelliptic curves of genus 3.
• Bertini involutions correspond bijectively to isomorphism classes of non-hyperelliptic curves of genus 4 whose canonical models lie on a singular quadric.
• All De Jonquières involutions of degree 2 are conjugate via linear automorphisms, forming a single conjugacy class.
• The normalized fixed curve of a De Jonquières involution of degree g+2 is a plane curve of degree g+2 with a single ordinary g-tuple point and no other singularities.