Hi!

I am looking for notions of general position that are stronger than linear general position.

To illustrate, 3 points in linear general position don't lie on a line. I want a notion that would claim that 6 points in general position don't lie on a quadric.

To be more specific: Assume I have an ideal $I$ in $\mathbb{C}[x_1,\dots,x_n]$ with a set of generators $p_1,\dots,p_k$. Assume the degrees of the generators are $d_1,\dots,d_k$ respectively, I would like to claim that no $f(n,d_1,\dots,d_k)$ points in general position are the common roots of $I$.

Restrictions $I$ can be made (for example it comes for a parametrized family etc.).

Is anyone familiar with a "general position" notion that captures this intuition? Can anyone refer me to a discussion on notions of "general position" that are close or related to such a notion?