I got myself in a confusing situation the other week while trying to read a bit of algebraic geometry. I'm hoping someone can pull me out.
Suppose $k$ is a field, and $V$ a homogeneous variety with generic point $(x)$ over $k$. Denote by $Z$ the algebraic set of zeroes in $k^a$ of a homogeneous ideal in $k[X]$ generated by forms $f_1,\dots,f_r\in k[X]$. It is a theorem of Hilbert and Zariski that $V\cap Z$ has only the trivial zero if and only if each $x_i$ is integral over $k[f_1(x),\dots,f_r(x)]$.
I searched for a proof of this fact and played with it over the course of last week, and came up with nothing. I would like to know, why does $V\cap Z$ have only the trivial zero if and only if each $x_i$ is integral over $k[f_1(x),\dots,f_r(x)]$?
Thank you for your expertise.