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?