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

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

2001-11-13

arxiv.org/abs/math/0111147v1

Mathematics

Algebraic Geometry

7 pages. AMSTEX

Scientific paper

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.

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