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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