Quadro-quadric special birational transformations of projective spaces

TL;DR

This paper classifies special birational transformations of projective spaces defined by quadrics, extending known classifications to broader classes of Fano manifolds and rational homogeneous varieties.

Contribution

It extends the classification of quadro-quadric transformations to include a wider class of prime Fano manifolds and rational homogeneous varieties.

Findings

Classification of transformations when Z is a Grassmannian, spinor, or E6 variety.

Extension of Ein and Shepherd-Barron's classification to broader Fano manifolds.

Complete classification of certain birational transformations onto rational homogeneous varieties.

Abstract

Special birational transformations $\Phi:\p^r\da Z$ 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 $\Phi:\p^r\da Z$ defined by quadric hypersurfaces onto (a linear setion of) a rational homogeneous variety different from a projective space and a quadric hypersurface.