Varieties with quadratic entry locus, I

Abstract

Quadratic entry locus manifold of type $\delta$ $X\subset\mathbb P^N$ of dimension $n\geq 1$ are smooth projective varieties such that the locus described on $X$ by the points spanning secant lines passing through a general point of the secant variety $SX\subseteq\mathbb P^N$ is a smooth quadric hypersurface of dimension $\delta=2n+1-\dim(SX)$ equal to the secant defect of $X$. These manifolds appear widely and naturally among projective varieties having special geometric properties and/or extremal tangential behaviour. We prove that, letting $\delta=2r_X +1\geq 3$ or $\delta=2r_X+2$, then $2^{r_X}$ divides $n-\delta$. This is obtained by the study of the projective geometry of the Hilbert scheme $Y_x\subset \mathbb(T_x^*)$ of lines passing through a general point $x$ of $X$, allowing an inductive procedure. The Divisibility Property described above allows unitary and simple proofs of many results on $QEL$-manifolds such as the complete classification of those of type $\delta\geq n/2$, of Cremona transformation of type $(2,3)$, $(2,5)$. In particular we propose a new and very short proof of the fact that Severi varieties have dimension 2,4, 8 or 16 and also an almost self contained half page proof of their classification due to Zak.