Varieties with quadratic entry locus, I

We introduce and study (L)QEL-manifolds X ⊂ P^N of type δ, a class of projective varieties whose extrinsic and intrinsic geometry is very rich, especially when δ > 0. We prove a strong Divisibility Property for LQEL-manifolds of type δ ≥ 3, allowing the classification of those of type δ ≥ dim(X)/2. In particular we obtain 2 a new and very short proof that Severi varieties have dimension 2,4, 8 or 16 and also an almost self-contained proof of their classification due to Zak. We also provide the classification of special Cremona transformations of type (2,3) and (2,5).