no. 1
Quadro-quadric Cremona transformations in low dimensions via the JC-correspondence
[Transformations de Crémona quadro-quadriques en basses dimensions via la correspondence JC]
Pirio, Luc1 ; Russo, Francesco2
1 IRMAR, UMR 6625 du CNRS, Université Rennes 1, Campus de beaulieu, 35000 Rennes, France
2 Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy
Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 71-111.

Il a été établit précédemment qu’une transformation de Crémona de bidegré (2,2) est linéairement équivalente à la projectivation de l’inversion d’une algèbre de Jordan de rang 3. Ce résultat (appelé la “correspondance JC”) est utilisé dans le présent article pour étudier les transformations birationnelles quadro-quadriques des espaces projectifs de petite dimension. En particulier, nous décrivons de nouvelles familles très simples de telles applications birationnelles et nous obtenons leur classifications complètes et explicites en dimension 4 et 5.

It has been previously established that a Cremona transformation of bidegree (2,2) is linearly equivalent to the projectivization of the inverse map of a rank 3 Jordan algebra. We call this result the “JC-correspondence”. In this article, we apply it to the study of quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension 4 and 5.

DOI : 10.5802/aif.2839
Classification : 14E07, 17Cxx
Keywords: Cremona transformation, Jordan algebra
Mot clés : Transformation de Crémona, Algèbre de Jordan

Pirio, Luc 1 ; Russo, Francesco 2

1 IRMAR, UMR 6625 du CNRS, Université Rennes 1, Campus de beaulieu, 35000 Rennes, France
2 Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy 
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Pirio, Luc; Russo, Francesco. Quadro-quadric Cremona transformations in low dimensions via the $JC$-correspondence. Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 71-111. doi : 10.5802/aif.2839. https://aif.centre-mersenne.org/articles/10.5802/aif.2839/

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