On the irreducibility of secant cones, and an application to linear normality
英文摘要：
Let $Y \subset ¶^r$ be a normal nondegenerate m-dimensional subvariety and let $σ(Y)$ 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 $¶^{r-1}$, proving that if the secant variety of $Y$ is the whole space and $σ(Y) < 2m - r + 1$, 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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