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**Birational involutions of \({\mathbb{P}}^ 2\).**
*(English)*
Zbl 1055.14012

Introduction: This paper is devoted to the classification of the elements of order 2 in the group \(\text{Bir}\,\mathbb{P}^2\) of birational automorphisms of \(\mathbb{P}^2\), up to conjugacy. This is a classical problem, which seems to have been considered first by Bertini. Bertini’s proof is generally considered as incomplete, as well as several other proofs which followed. We refer to the introduction of G. Castelnuovo and F. Enriques [Palermo Rend. 14, 290–302 (1900; JFM 31.0658.02)] for a more detailed story and for an acceptable proof. However the result itself, as stated by these authors, is not fully satisfactory: Since they do not exclude singular fixed curves, their classification is somewhat redundant.

We propose in this paper a different approach, which provides a precise and complete classification. It is based on the simple observation that any birational involution of \(\mathbb{P}^2\) is conjugate, via an appropriate birational isomorphism \(S@>\sim>>\mathbb{P}^2\), to a biregular involution \(\sigma\) of a rational surface \(S\). We are thus reduced to the birational classification of the pairs \((S,\sigma)\), a problem very similar to the birational classification of real surfaces. This classification has been achieved by classical geometers; the case of surfaces with a finite group of automorphisms has been treated more recently along the same lines by Yu. I. Manin [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 133, 160–176 (1984; Zbl 0542.53039)].

These questions have been greatly simplified in the early 80’s by the introduction of Mori theory. In our case a direct application of this theory shows that the minimal pairs (\(S,\sigma)\) fall into two categories, those which admit a \(\sigma\)-invariant base-point free pencil of rational curves, and those with \(\text{rk\,Pic} (S)^\sigma=1\). The first case leads to the so-called De Jonquiéres involutions in the second case an easy lattice-theoretic argument shows that the only new possibilities are the celebrated Geiser and Bertini involutions. Any birational involution is therefore conjugate to one (and only one) of these three types.

We propose in this paper a different approach, which provides a precise and complete classification. It is based on the simple observation that any birational involution of \(\mathbb{P}^2\) is conjugate, via an appropriate birational isomorphism \(S@>\sim>>\mathbb{P}^2\), to a biregular involution \(\sigma\) of a rational surface \(S\). We are thus reduced to the birational classification of the pairs \((S,\sigma)\), a problem very similar to the birational classification of real surfaces. This classification has been achieved by classical geometers; the case of surfaces with a finite group of automorphisms has been treated more recently along the same lines by Yu. I. Manin [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 133, 160–176 (1984; Zbl 0542.53039)].

These questions have been greatly simplified in the early 80’s by the introduction of Mori theory. In our case a direct application of this theory shows that the minimal pairs (\(S,\sigma)\) fall into two categories, those which admit a \(\sigma\)-invariant base-point free pencil of rational curves, and those with \(\text{rk\,Pic} (S)^\sigma=1\). The first case leads to the so-called De Jonquiéres involutions in the second case an easy lattice-theoretic argument shows that the only new possibilities are the celebrated Geiser and Bertini involutions. Any birational involution is therefore conjugate to one (and only one) of these three types.

### MSC:

14E07 | Birational automorphisms, Cremona group and generalizations |

14N05 | Projective techniques in algebraic geometry |

14E05 | Rational and birational maps |

14E30 | Minimal model program (Mori theory, extremal rays) |