Marci Hablicsek (a finishing Ph.D. student at UW) and I recently posted a new preprint, “An incidence conjecture of Bourgain over fields of finite characteristic.”

The theme of the paper is a beautiful theorem of Larry Guth and Nets Katz, one of the early successes of Dvir’s “polynomial method.” They proved a conjecture of Bourgain:

Given a set S of points in R^3, and a set of N^2 lines such that

- No more than N lines are contained in any plane;
- Each line contains at least N points of S;
then S has at least cN^3 points.

In other words, the only way for a big family of lines to have lots of multiple intersections is for all those lines to be contained in a plane. (In the worst case where *all* the lines are in a plane, the incidences between points and lines are governed by the Szemeredi-Trotter theorem.)

I saw Nets speak about this in Wisconsin, and I was puzzled by the fact that the theorem only applied to fields of characteristic 0, when the proof was entirely algebraic. But you know the proof must fail somehow in characteristic p, because the statement isn’t true in characteristic p. For example, over the field k with p^2 elements, one can check that the *Heisenberg surface*

has a set of p^4 lines, no more than p lying on any plane, and such that each line contains at least p^2 elements of X(k). If the Guth-Katz theorem were true over k, we could take N = p^2 and conclude that |X(k)| is at least p^6. But in fact, it’s around p^5.

It turns out that there is one little nugget in the proof of Guth-Katz which is not purely algebraic. Namely: they show that a lot of the lines are contained in some surface S with the following property; at every smooth point s of S, the tangent plane to S at s intersects S with multiplicity greater than 2. They express this in the form of an assertion that a certain curvature form vanishes everywhere. In characteristic 0, this implies that S is a plane. But not so in characteristic p! (As always, the fundamental issue is that a function in characteristic p can have zero derivative without being constant — viz., x^p.) All of us who did the problems in Hartshorne know about the smooth plane curve over F_3 with every point an inflection point. Well, there are surfaces like that too (the Heisenberg surface is one such) and the point of the new paper is to deal with them. In fact, we show that the Guth-Katz theorem is true word for word as long as you prevent lines not only from piling up in planes but also from piling up in these “flexy” surfaces.

It turns out that any such surface must have degree at least p, and this enables us to show that the Guth-Katz theorem is actually true, word for word, over the prime field F_p.

If you like, you can think of this as a strengthening of Dvir’s theorem for the case of F_p^3. Dvir proves that a set of p^2 lines with no two lines in the same direction fills up a positive-density subset of the whole space. What we prove is that the p^2 lines don’t have to point in distinct directions; it is enough to impose the weaker condition that no more than p of them lie in any plane; this already implies that the union of the lines has positive density. Again, this strengthening doesn’t hold for larger finite fields, thanks to the Heisenberg surface and its variants.

This is rather satisfying, in that there are other situations in this area (e.g. sum-product problems) where there are qualitatively different bounds depending on whether the field k in question has nontrivial subfields or not. But it is hard to see how a purely algebraic argument can “see the difference” between F_p and F_{p^2}. The argument in this paper shows there’s at least one way this can happen.

Satisfying, also, because it represents an unexpected application for some funky characteristic-p algebraic geometry! I have certainly never needed to remember that particular Hartshorne problem in my life up to now.