A Note on Generic Projections

Abstract

Let $X \subseteq {\bf P}^N ={\bf P}^{2n}_K$ be a subvariety of dimension $n$ and $P \in {\bf P}^N$ a generic point. If the tangent variety Tan$ X$ is equal to ${\bf P}^N$ then for generic points $x$, $y$ of $X$ the projective tangent spaces $t_xX$ and $t_yX$ 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 $R(π_P)$ of the linear projection $π_P:X\to {\bf P}^{N-1}$.