A Note on Generic Projections

Abstract

Let $X\subseteq {\mathbf{P}}^N={\mathbf{P}}_K^{2n}$ be a subvariety of dimension $n$ and $P\in {\mathbf{P}}^N$ a generic point. If the tangent variety Tan$X$ is equal to ${\mathbf{P}}^N$ then for generic points $x$, $y$ of $X$ the projective tangent spaces $t_{x}X$ and $t_{y}X$ meet in one point $P=P(x,y)$. The main result of this paper is that the rational map $(x,y)\mapsto P(x,y)$ is dominant. In other words, a generic point $P$ is uniquely determined by the ramification locus $R({\pi }_P)$ of the linear projection ${\pi }_P:X\to {\mathbf{P}}^{N-1}$.