Incidences between points and curves with almost two degrees of freedom

Abstract

We study incidences between points and (constant-degree algebraic) curves in three dimensions, taken from a family C of curves that have almost two degrees of freedom, meaning that (i) every pair of curves of C intersect in O(1) points, (ii) for any pair of points p, q, there are only O(1) curves of C that pass through both points, and (iii) there exists a 6-variate real polynomial F of constant degree, so that a pair p, q of points admit a curve of C that passes through both of them if and only if F(p,q)=0. (As an example, the family of unit circles in R3 that pass through some fixed point is such a family.)We begin by studying two specific instances of this scenario. The first instance deals with the case of unit circles in R3 that pass through some fixed point (so called anchored unit circles). In the second case we consider tangencies between directed points and circles in the plane, where a directed point is a pair (p,u), where p is a point in the plane and u is a direction, and (p,u) is tangent to a circle γ if p∈γ and u is the direction of the tangent to γ at p. A lifting transformation due to Ellenberg et al. [3] maps these tangencies to incidences between points and curves (‘lifted circles’) in three dimensions. In both instances we have a family of curves in R3 with almost two degrees of freedom.We show that the number of incidences between m points and n anchored unit circles in R3, as well as the number of tangencies between m directed points and n arbitrary circles in the plane, is O(m3/5n3/5+m+n).We then derive a similar incidence bound, with a few additional terms, for more general families C of curves in R3 with almost two degrees of freedom, under a few additional natural assumptions.The proofs follow standard techniques, based on polynomial partitioning, but they face a critical novel issue involving the analysis of surfaces that are infinitely ruled by the respective family of curves, as well as of surfaces in a dual three-dimensional space that are infinitely ruled by the respective family of suitably defined dual curves. We either show that no such surfaces exist, or develop and adapt techniques for handling incidences on such surfaces.The general bound that we obtain isI(P,C)=O(m3/5n3/5+(m11/15n2/5+n8/9)δ1/3+m2/3n1/3π1/3+m+n), where π (resp., δ) is the maximal number of curves (resp., dual curves) that lie on a common surface that is infinitely ruled by the family of these curves.