Varieties with Quadratic Entry Locus, I

We shall introduce and study quadratic entry locus varieties, a class of projective algebraic varieties whose extrinsic and intrinsic geometry is very rich. Let us recall that, for an irreducible non-degenerate variety X ⊂ P of dimension n ≥ 1, the secant defect of X , denoted by δ(X), is the difference between the expected dimension and the effective dimension of the secant variety SX ⊆ P of X , that is δ(X) = 2n+ 1 − dim(SX). This is an important projective invariant measuring the dimension of the entry locus Σp ⊆ X described by the points on X spanning secant lines passing through a general point p ∈ SX ; see Section 1. Many examples appearing in different settings suggested the definition of quadratic entry locus manifold of type δ, briefly QEL-manifold of type δ. These are smooth varieties X ⊂ P for which Σp is a smooth quadric hypersurface of dimension δ = δ(X), whose linear span in P is the locus of secant lines to X passing through p, p ∈ SX general; see Section 1. When δ = 0 and N > 2n + 1 this condition does not impose strong restrictions on X . On the contrary QEL-manifolds X ⊂ P of type δ = 0 are linearly normal and rational, see [CMR], while in [IR2] it is proved that every QEL-manifold of type δ > 0 is rational. The notion of QEL-manifold was also motivated by the remark that a lot of secant defective smooth varieties with special geometric properties and/or with extremal tangential behaviour are QEL-manifolds: varieties defined by quadratic equations having enough linear syzygies, for example satisfying condition N2 of Green (Proposition 1.4); homogeneous varieties, secant defective or not; Scorza Varieties (and in particular Severi Varieties; this property being an essential ingredient for their classification, cf. [Z2] and Section 3); centers of special Cremona transformations of type (2, d) ([ESB] and Section 4), varieties whose dual variety is small ([E1] and [IR2]) and which are not hypersurfaces. Furthermore, if n = 2, 3, any smooth secant defective variety X ⊂ P with SX ( P is a QEL-manifold; see [Se, S1, F1]. By definition QEL-manifolds of type δ > 0, or their isomorphic projections, form very special examples of rationally connected varieties [KMM, Ko, De]. Gaetano Scorza was the first who realized the link between secant defective varieties and rational connectedness in the pioneering papers [S1, S3], where, ante litteram, the condition of rational connectedness by conics appears for le varietá della (n − 1)-esima specie, o dell’ ultima specie in [S3, pp. 252–253]. These definitions are explained in detail in [Ru] and also inspired some results in [IR1] and in [IR2]. Here we develop the theory of QEL-manifolds of type δ ≥ 1 by studying the geometry of the family of conics and, for δ ≥ 2, of lines produced by the quadratic entry loci via the modern tools of deformation theory of rational curves on a manifold and via their parameter spaces. For δ ≥ 2, the lines passing through a general point x ∈ X describe a smooth, not necessarily irreducible, variety Yx ⊂ P((Tx,X)) = P, defined in Proposition 2.2. The most important results for QEL-manifolds of type δ ≥ 2 are consequences of the study of the projective geometry of the subvariety Yx ⊂ P, especially for δ ≥ 3 when Yx is also irreducible. This is not surprising since the family of lines on such an X ⊂ P is the minimal covering family of rational curves in the sense of