Quadro-quadric special birational transformations of projective spaces

Abstract

Special birational transformations $Φ:\p^r\da Z$ defined by quadric hypersurfaces are studied by means of the variety of lines $\mathcal L_z\subset\p^{r-1}$ passing through a general point $z\in Z$. Classification results are obtained when $Z$ is either a Grassmannian of lines, or the 10-dimensional spinor variety, or the $E_6$-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 $Φ:\p^r\da Z$ defined by quadric hypersurfaces onto (a linear setion of) a rational homogeneous variety different from a projective space and a quadric hypersurface.

First draft

References in corpus (5)

• Varieties with quadratic entry locus, I
• Manifolds covered by lines and the Hartshorne Conjecture for quadratic manifolds
• On special quadratic birational transformations of a projective space into a hypersurface
• On special quadratic birational transformations whose base locus has dimension at most three
• Special birational transformations of projective spaces

Cited by in corpus (5)

• Examples of special quadratic birational transformations into complete intersections of quadrics
• Special birational transformations of projective spaces
• Quadro-quadric special birational transformations from projective spaces to smooth complete intersections
• Special cubic birational transformations of projective spaces
• Special birational transformations of type (2,1)