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

Abstract

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$ofitsgenericprojectionto$¶^{r-1}$,provingthatifthesecantvarietyof$Y$isthewholespaceand$σ(Y) < 2m - r + 1$,then$D$isirreducible.ApplyingZa{k}^{\prime }sTangencytheoremwededucetheirreducibilityof$D$when$m > 2(r-1)/3$. The latter implies a version of Zak's Linear Normality theorem.

7 pages. AMSTEX