Quadro-Quadric Special Birational Transformations of Projective Spaces

Special birational transformations $\phi:\mathbb P^r\longrightarrow Z$ defined by quadric hypersurfaces are studied by means of the variety of lines $\mathcal L_Z\subset\mathbb P^{r-1}$ passing through a general point z∈Z. Classification results are obtained when Z is either a Grassmannian of lines, or the 10-dimensional spinor variety, or the E6-variety. In the particular case of quadro-quadric transformations, we extend the well-known classification of Ein and Shepherd-Barron coming from Zak's classification of Severi varieties to a wider class of prime Fano manifolds Z. Combining both results, we get a classification of special birational transformations $\phi:\mathbb P^r\longrightarrow Z$ defined by quadric hypersurfaces onto (a linear section of) a rational homogeneous variety different from a projective space and a quadric hypersurface.