I have a question about points in general position. In my mind I am imagining that I am working over the field $\mathbb{C}$ and in dimension $d=3$, but I am stating the problem with slightly more generality.
Suppose I have a fixed family $\mathcal{F}$ of polynomials $P(X,Y)$ defined on $(\mathbb{F}^d)^n\times(\mathbb{F}^d)^m$ (so $X\in (\mathbb{F}^d)^n$ is interpreted as a set of $n$ points in $\mathbb{F}^d$ and likewise $Y$ is a set of $m$ points).
In my particular case the family $\mathcal{F}$ is a special subset of quadratic polynomials with coefficients in $\{-1,0,1\}$.
I will say that a subset of polynomials $S\subset\mathcal{F}$ is admissible if there exists a configuration $X\in(\mathbb{F}^d)^n$ which is "generic with respect to $\mathcal{F}$" and any other configuration $Y\in(\mathbb{F}^d)^m$ such that $P(X,Y)=0$ for all $P\in S$.
Question: If $S$ is admissible and $X'$ is any configuration of points, does there exist $Y'$ such that $P(X',Y')=0$ for all $P\in S$?
My intuition is that any coincidences that allowed a configuration $Y$ such that $P(X,Y)=0$ must have nothing to do with the special arrangement of $X$, and must therefore persist for other configurations. So what I am really asking is whether there exists a reasonable notion of "generic with respect to $\mathcal{F}$" for which the above question has a positive answer.