Cremona transformations of \(\mathbb P^4\) coming from complete intersection of quadrics
The Cremona group \(\mathrm{Bir}(\mathbb{P}^n)\) is the group of birational selfmaps of the projective space. In dimension 2 the group \(\mathrm{Bir}(\mathbb{P}^2)\) is well known, while in higher dimension many important questions are still open. In the paper under review a family of elements of \(\mathrm{Bir}(\mathbb{P}^4)\) is described. It is obtained in the following way: Let \(\alpha_1,\alpha_2\subseteq \mathbb{P}^7\) be two planes and \(Q_1, Q_2, Q_3\) quadric hypersurfaces such that \(X:=Q_1\cap Q_2\cap Q_3\) is smooth and contains \(\alpha_1\) and \(\alpha_2\). Let \(\pi_i\) be the restriction to \(X\) of the projection from \(\alpha_i\). Then \(\pi_1\circ\pi_2^{-1}\) induces an element of \(\mathrm{Bir}(\mathbb{P}^4)\). The authors prove that this element is a determinantal Cremona transformation if \(\alpha_1\cap\alpha_2=\emptyset\), it is de Jonquières if \(\alpha_1\cap\alpha_2\) is a line and it is neither determinantal nor de Jonquières if \(\alpha_1\cap\alpha_2\) is a point.
scientific article; zbMATH DE number 6858597
scientific article