Varieties with quadratic entry locus, I

Abstract

Quadratic entry locus manifold of type $\delta$ $X\subset {\mathbb{P}}^N$ of dimension $n\ge 1$ 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 $\delta =2r_X+1\ge 3$ 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 $Y_x\subset (T_x^{\ast })$ 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 $\delta \ge n/2$, 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.

16 pages; some misprints and imprecisions corrected; some references added; final version as appeared in Math. Ann