I was hoping to ask a small follow up to the question I asked here.
Suppose $V$ is an algebraic variety over arbitrary field $k$. (For this situation, I'll take the definition $\dim\ V=\deg_k(k(x))$, where $(x)=(x_1,\dots,x_n)\in V$ is a generic point, and by $\deg$ I mean the transcendence degree.) As usual, $V(f_1,\dots,f_s)$ is the set of zeroes of the homogeneous forms $f_1,\dots,f_s$ in the affine space.
Now say you take $U$ to be an algebraic set $x_1=\cdots=x_p=0$, (so $U$ is the algebraic set with associated ideal $(x_1,\dots,x_p)$) that is the algebraic set of coordinates in $\mathbb{A}^n$ where the first $p$ coordinates are $0$, and where $p<\dim\ V$. Is it now the case that $U\cap V\neq\{0\}$? Many thanks.