Extraordinary Hypersurfaces in Positive Characteristic

Steven Kleiman
Massachusetts Institute of Technology

We consider some algebraic geometry associated with the recent work of Ellenberg and Hablicsek on the incidence conjecture of Bourgain over a field k of positive charateristic. Their result asserts this: given N^2 lines in k^3 and a set S of points, there's a universal constant c such that |S| > cN^3 if each line contains at least N of the points and no more than 2Nd lines lie in any flexy surface of degree d.

By definition, a FLEXY surface is a surface X such that, at every simple point x, the tangent plane T_x meets X in a curve with multiplicity at least 3. We discuss the shape of the equation of X and the geometry of its Gauss map, x |-> T_x. In particular, we consider a class of particularly beautiful flexy surfaces, the Hermitian surfaces, which have only about N^{5/2} points, yet carry about N^2 lines, each with at least N points.

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