Puzzle: low-height points in general position

I have no direct reason to need the answer to, but have wondered about, the following question.

We say a set of points in are in general position if the Hilbert function of any subset S of the points is equal to the Hilbert function of a generic set of points in .  In other words, there are no curves which contain more of the points than a curve of their degree “ought” to.  No three lie on a line, no six on a conic, etc.

Anyway, here’s a question.  Let H(N) be the minimum, over all N-tuples  of points in general position, of

where H denotes Weil height.  What are the asymptotics of H(N)?  If you take the N lowest-height points, you will have lots of collinearity, coconicity, etc.  Does the Bombieri-Pila / Heath-Brown method say anything here?