The set of cubic hypersurfaces in $\mathbb{P}^3$ are paramatrized by a 19-dimensional projective space. If we consider only the hypersurfaces containing a fixed generic set of 19 points, it seems to me we can expect to end up with a finite set. Do we know how large this set is?
More generally, how many degree $d$ hypersurfaces in $\mathbb{P}^n$ contain a generic set of $\binom{d+n}{d}-1$ points?