On the irreducibility of secant cones, and an application to linear normality

Let $Y \subset \P^r$ be a normal nondegenerate m-dimensional subvariety and let $\sigma(Y)$ denote the maximum dimension of a subvariety $Z \subset Y_{smooth}$ such that $Z$ contains a generic point of some divisor on $Y$ and the tangent planes $T_y Y$ for all $y \in Z$ are contained in a fixed hyperplane. In this article we study the double locus $D \subset$Y$of its generic projection to$\P^{r-1}$, proving that if the secant variety of$Y$is the whole space and$\sigma(Y) < 2m - r + 1$, then$D$is irreducible. Applying Zak's Tangency theorem we deduce the irreducibility of$D$when$m > 2(r-1)/3$. The latter implies a version of Zak's Linear Normality theorem.