Strong notions of general position 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?
 A: The defining equation of a degree $d$ hypersurface in $\mathbb P^n$ has $n+d\choose d$ coefficients and hence these hypersurfaces may be parametrized by a projective space of dimension ${n+d\choose d} -1$. Picking a point to be contained by the hypersurface is a linear equation on the coefficients of these defining equations so any set of ${n+d\choose d} -1$ points is contained in at least one such hypersurface, but a set of $n+d\choose d$ points in general position is not. 
If you take $n=2$ and $d=1$ you get that $3$ points in general position do not lie on a line, if you take $d=2$, then you get that $6$ points in general position do not lie on a quadric, etc.
As far as the definition of "in general position" goes, in this situation, one could say that a set of points is in general position if they impose independent conditions on the coefficients of the defining equations.
More generally, you can consider any parameter space of any type of subobjects of any fixed variety/scheme/space. Containing a point of this fixed space will impose a condition on the parameter space. You can define points being in general position with respect to the type of subobjects you're considering by requiring that the conditions they impose on the parameter space are independent.
A: If there is a family of objects parametrized by points $a$ in some topological space $X$
they usually say that "an object is in general position" (or "generic") to mean that $a$ belongs to an open dense subset of $X$. This notion of course depends on topology on $X$. When $X$ is an algebraic variety, say over $R$ or $C$, one can consider two natural topologies: the usual one and Zariski one.
I think this applies to all examples given in the question and in the answer of Sandor.
A: As I indicated in my comment, if you're looking for an intuitive and easy to state strong notion of general position, you might ask that the coordinates of the points are algebraically independent over the rationals (that is, there is no non-zero rational-coefficient polynomial in the coordinates that vanishes). This notion is used, for example in this paper on generic rigidity. This is a pretty arbitrary condition and may be unnecessarily strong, but it is also very powerful and could effectively capture the intuitive idea of a "generic" set of points.
