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 “-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 ”) 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.
Accepted:
DOI: 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}, pages = {71--111}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {64}, number = {1}, year = {2014}, doi = {10.5802/aif.2839}, zbl = {06387266}, mrnumber = {3330541}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2839/} }
TY - JOUR TI - Quadro-quadric Cremona transformations in low dimensions via the $JC$-correspondence JO - Annales de l'Institut Fourier PY - 2014 DA - 2014/// SP - 71 EP - 111 VL - 64 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2839/ UR - https://zbmath.org/?q=an%3A06387266 UR - https://www.ams.org/mathscinet-getitem?mr=3330541 UR - https://doi.org/10.5802/aif.2839 DO - 10.5802/aif.2839 LA - en ID - AIF_2014__64_1_71_0 ER -
%0 Journal Article %T Quadro-quadric Cremona transformations in low dimensions via the $JC$-correspondence %J Annales de l'Institut Fourier %D 2014 %P 71-111 %V 64 %N 1 %I Association des Annales de l’institut Fourier %U https://doi.org/10.5802/aif.2839 %R 10.5802/aif.2839 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2839/
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