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

On the dimension of secant varieties

“…A quick proof of the classification of Severi varieties can be obtained by using the beautiful ideas contained in [21]. Along the same lines one can give a proof of the classification of k-Severi varieties, alternative to the one described above based on Zak's classification of Scorza varieties.…”

Inductive characterizations of hyperquadrics

“…[7,9,5]) a smooth irreducible non-degenerate variety X ⊂ P N is said to be a local quadratic entry locus manifold of type δ ≥ 0 (LQEL-manifold for short) if for general x, y ∈ X distinct points, there exists a hyperquadric of dimension δ = δ(X ) contained in X and passing through x, y. By definition, a LQEL-manifold of positive secant defect is conically connected, but the converse is not true.…”

“…A systematic study of LQEL-manifolds has been successively carried out by Russo in [9], in particular, the following remarkable theorem has been proved in [9]. Theorem 1 ([9, Theorem 2.8]) For an n-dimensional LQEL-manifold X ⊂ P N of type δ ≥ 3, let x ∈ X be a general point and let Y x ⊂ P n−1 be the Hilbert scheme of lines on X passing through x.…”

“…This gives a substantial improvement to the Main Theorem 0.2 of [7], where the same claim is proved under the assumption that a general hyperquadric in the family is smooth and that m ≥ 3n/5 + 1. Our proof here, based on ideas contained in [5] and [9], is much simpler and is completely different from that in [7]. However, we should point out that a more general result, without assuming the quadric subspaces pass all through a fixed point, is proven in [7].…”

“…The same idea of proof, combined with the Divisibility Theorem of [9], allows us to prove (cf. Corollary 3) that for an n-dimensional LQEL-manifold, either it is a hyperquadric or its secant defect is no bigger than n+8 3 .…”