On the Irreducibility of Secant Cones, and an Application to Linear Normality

Given a smooth subvariety of dimension > 2 3 (r − 1) in Pr, we show that the double locus (upstairs) of its generic projection to P is irreducible. This implies a version of Zak’s Linear Normality theorem. A classical, and recently revisited (cf. [GP, L, Pi] and references therein), method for studying the geometry of a subvariety Y in P is to project Y generically to a lower-dimensional projective space, for example so that Y maps birationally to a (singular) hypersurface Ȳ ⊂ P. To make use of this method, it is usually important to have precise control over the singularities of Ȳ and in particular over the entire singular (=double) locus DY of Ȳ and its inverse image CY in Y . As the dimension of these is easily determined, a natural question is: are CY and DY irreducible? This question plays an important role, for instance, in Pinkham’s work on regularity bounds for surfaces [Pi]. The purpose of this note is to show that this irreducibility holds provided the codimension of Y is sufficiently small compared to its dimension (see Theorems 1,2 and Corollary 3 below). As an application we give a proof of Zak’s linear normality theorem (in a slightly restricted range, see Corollary 4 below). Indeed the results seem closely related as our argument ultimately depends on having a bound on the dimension of singular loci of hyperplane sections, manifested in the form of the integer σ(Y ) (see Thm. 1 below), and it is Zak’s theorem on tangenciesalso a principal ingredient in other proofs of linear normalitythat gives us good control over σ(Y ). We begin with some definitions. Let Y denote an irreducible m− dimensional subvariety of P. As usual, we mean by a secant line of Y a limit of lines in P spanned by pairs of distinct points of Y . The union of all secant lines is denoted by Sec(Y ). Y is said to be projectable if Sec(Y ) ( P. For any linear subspace Q ⊂ P, we let