Algebraic curves, rich points, and doubly-ruled surfaces

Abstract

We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let k beafieldandlet L be a collection of n space curves in k3, with n ≪ (char(k))2 or char(k) =0. Then either (a) there are at most O(n3/2) points in k3 hit by at least two curves, or (b) at least Ω(n1/2 ) curves from L must lie on a bounded-degree surface, and many of the curves must form two “rulings” of this surface. We also develop several new tools including a generalization of the classical flecnode polynomial of Salmon and new algebraic techniques for dealing with this generalized flecnode polynomial.