Quadro-quadric Cremona transformations in low dimensions via the JC-correspondence
Annales de l'Institut Fourier, Volume 64 (2014) no. 1, p. 71-111

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.

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.

DOI : https://doi.org/10.5802/aif.2839
Classification:  14E07,  17Cxx
Keywords: Cremona transformation, Jordan algebra
@article{AIF_2014__64_1_71_0,
     author = {Pirio, Luc and Russo, Francesco},
     title = {Quadro-quadric Cremona transformations in low dimensions via~the~$JC$-correspondence},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {1},
     year = {2014},
     pages = {71-111},
     doi = {10.5802/aif.2839},
     mrnumber = {3330541},
     zbl = {06387266},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2014__64_1_71_0}
}
Pirio, Luc; Russo, Francesco. Quadro-quadric Cremona transformations in low dimensions via the $JC$-correspondence. Annales de l'Institut Fourier, Volume 64 (2014) no. 1, pp. 71-111. doi : 10.5802/aif.2839. http://www.numdam.org/item/AIF_2014__64_1_71_0/

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