Cremona transformations of $${\mathbb P}^4$$ P 4 coming from complete intersection of quadrics

Abstract

We consider Cremona transformations $$\phi {:}\,{\mathbb P}^4\dashrightarrow {\mathbb P}^4$$ which factorize through projections of a smooth complete intersection of quadrics in $${\mathbb P}^7$$ . We prove there are three types of such transformations according to the relative position of the centers of projection. Moreover, in order to fix one of these three types, we give a geometric characterization of Cremona transformations of $${\mathbb P}^n$$ which act birationally on the set of hyperplanes passing through a point.