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Quadro-quadric Cremona transformations in low dimensions via the  J C -correspondence

Luc Pirio, Francesco Russo (2014)

Annales de l’institut Fourier

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 “ J C -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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