Incidences Between Points and Curves with Almost Two Degrees of Freedom

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) 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 for some polynomial F of constant degree associated with the problem. (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. 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(m^(3/5)n^(3/5)+m+n) in both cases. We then derive a similar incidence bound, with a few additional terms, for more general families 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 is O(m^(3/5)n^(3/5)+m+n) plus additional terms that depend on how many curves or dual curves can lie on an infinitely-ruled surface.