Quadro-quadric special birational transformations from projective spaces to smooth complete intersections (Q2790323)

Let \(Z\subset \mathbb P^N\) be a prime Fano manifold of dimension \(r\), covered by lines. A birational map \(\phi: \mathbb P^r\dashrightarrow Z\) is called a special birational transformation if the base locus scheme of \(\phi\) is smooth and connected. It is called quadro-quadric if it is defined by a linear system of quadrics in \(\mathbb P^r\) and similarly \(\phi^{-1}\) is defined by a linear subsystem of \(|\mathcal O_Z(2)|\).NEWLINENEWLINEThe case \(Z=\mathbb P^r\) has been studied by Ein and Shepherd-Barron, who proved that the base loci of \(\phi\) and \(\phi^{-1}\) are both Severi varieties [\textit{L. Ein} and \textit{N. Shepherd-Barron}, Am. J. Math. 111, No. 5, 783--800 (1989; Zbl 0708.14009)]. \textit{A. Alzati} and \textit{J. C. Sierra} gave a complete classification when \(Z\) is a LQEL-manifold, i.e. a locally quadratic entry locus manifold [Adv. Math. 289, 567--602 (2016; Zbl 1358.14017)].NEWLINENEWLINEThe main results of the article under review are as follows. If \(\phi\) is a quadro-quadric special birational transformation as above, such that \(Z\) is a smooth non-degenerate hypersurface in \(\mathbb P^{r+1}\), then \(Z\) is a quadric hypersurface, the base locus of \(\phi^{-1}\) is a Severi variety, and the base locus of \(\phi\) is a hyperplane section of a Severi variety. In the same situation, if \(Z\) is a smooth non-degenerate complete intersection in \(\mathbb P^N\), then the same conclusion follows.NEWLINENEWLINEThese results complete previous work of \textit{G. Staglianò} [Rend. Circ. Mat. Palermo (2) 61, No. 3, 403--429 (2012; Zbl 1261.14005)].NEWLINENEWLINEThe proofs mainly rely on the study of the variety of minimal tangents \(Z^{(1)}\) of \(Z\) and on the Divisibility Theorem for QEL-manifolds due to Francesco Russo.