Quadro-quadric Cremona transformations in low dimensions via the $JC$-correspondence

Pirio, Luc; Russo, Francesco

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.

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