, Volume 344, Issue 3, pp 597-617

Varieties with quadratic entry locus, I

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Abstract

We introduce and study (L)QEL-manifolds \({X \subset \mathbb P^N}\) 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 \({\delta \geq \frac{dim(X)}{2}}\) . In particular we obtain 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).

Partially supported by CNPq (Centro Nacional de Pesquisa), grant 308745/2006-0, and by PRONEX/FAPERJ–Algebra Comutativa e Geometria Algebrica.