Special birational transformations of projective spaces

Quadro-Quadric Special Birational Transformations of Projective Spaces

“…In [1], we studied the case in which Z is a manifold with cyclic Picard group generated by the hyperplane section O Z (1) of Z ⊂ P α . We obtained a complete classification for n ∈ {1, 2}, m = {0, 1, 2} and r = n + 2, as well as some partial results for n = 3.…”

“…In this paper, we consider special birational transformations defined by quadric hypersurfaces. In this case, the techniques involved are different from those present in [1]. This is due to a couple of facts:…”

“…We deal with birational transformations Φ : P r Z from the complex projective space onto a prime Fano manifold whose base locus X ⊂ P r is a smooth irreducible and reduced scheme of dimension n. According to the classical terminology of Cremona transformations, these birational transformations are called special and they have been recently studied in [1], extending some of the main results of [2], [5] and [3]. In particular, complete classifications have been obtained in the cases n = 1, n = 2 and r − n = 2.…”

“…This being said, assume now that Z is a prime Fano manifold different from P r . A bunch of examples of special birational transformations of type (2,1) is given by the orbit of the highest weight vector of an irreducible representation of some algebraic groups, and is described in detail in [32, Ch. III, Theorem 3.8].…”

“…On the contrary, very few examples of special birational transformations of type (2, b) are known for b ≥ 3 (cf. [1], [28] and [29]).…”