## 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 $P_1, \ldots, P_N$ in $\mathbf{A}^2$ 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 $|S|$ points in $\mathbf{A}^n$.  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 $P_1, \ldots, P_N \in \mathbf{A}^2(\mathbf{Q})$ of points in general position, of

$\max H(P_i)$

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?

## 4 thoughts on “Puzzle: low-height points in general position”

1. jlk says:

One question. Which projective embedding are you using to define the Weil height?

2. jlk says:

Sorry! I think I misread your question. I thought you are taking the height of {P_1, …, P_N} as a point on the Hilbert scheme, but now I see I misunderstood the question.

3. D. Eppstein says:

Have you changed your wordpress software recently? For the last several days, every time my RSS newsreader checks your feed, it thinks all your postings from the last several months are new and shows them to me again. It’s only you and a couple of other wordpress blogs doing this, so I think the problem is more likely at your end than mine.

4. JSE says:

Nope, haven’t made any changes on this end. Readers, anybody else having this problem?