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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?

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Demanding that a cubic surface $ \sum (a_{ijk\ell}) x^iy^jz^kw^\ell $ pass through a specific point $(x_0: y_0: z_0: w_0)$ imposes a linear condition on the coefficients $a_{ijk\ell}$, which are the homogeneous coordinates of your $\mathbb{P}^{19}$. So you are taking the intersection of 19 hyperplanes in $\mathbb{P}^{19}$, which generically will be a point but may be infinite if you pick your points badly (it will never have a finite cardinality other than one, nor will it be empty). Same argument in the general case.

See

http://en.wikipedia.org/wiki/Five_points_determine_a_conic

(which is more general than the title suggests) for more detail.