Number of points determining a Quadric

Question

I know that in $\mathbb{R}^2$ that 5 points in general linear position determine a unique conic (also non-degenerate). I was wondering about the higher dimensional analogue of this. Is it true, for example, that 9 points in general linear position determine a unique quadric (also nondegenerate). The last statement in parenthesis seems a little wonky to me, because it seems like you can choose 9 points on a cylinder, without any 4 lying on the same plane. My question is how/what is the appropriate generalization of 5 points determining a conic, and how does one exclude degenerate cases (that is $n$ points satisfying some independence relation in $\mathbb{R}^m$ determine a unique quadric, and the quadric must be non-degenerate).

Answer 1

Yes, what you say is true, as long as you are clear about the meaning of "degenerate".

Forgetting for the time being about the problem of not having enough points, the basic point is that there are two equivalent ways to think of degenerate conics in the plane:

1) those which have a singular point (perhaps a "point at infinity"); or

2) those which are defined by a reducible form.

Now, when we pass to three dimensions, both of these notions still make sense, but they are no longer the same! In your example, a cylinder is degenerate in sense 1) above (in fact, in the world of projective varieties, it's the same as a cone) but it is not degenerate in the sense of 2).

So the correct statement will be that 9 points in $\mathbf{R}^3$ determine a quadric, and if those points are in linearly general position, then the quadric is not reducible, i.e. degenerate in sense 2) above.

If you want to make sure they don't lie on a quadric which is degenerate in sense 1), you have to be a little bit more picky; unfortunately at the moment I can't think of a nice geometric characterisation of these sets of points at the moment.

Finally, note that all these things generalise to higher dimensions: the general statement is that ${n+2 \choose 2} -1$ points in general position will determine a unique quadric.

Hope that helps!