On the use of the Klein quadric for geometric incidence problems in two dimensions

We discuss a unified approach to a class of geometric combinatorics incidence problems in two dimensions, of the Erdös distance type. The goal is obtaining the second moment estimate. That is, given a finite point set $S$ in two dimensions, and a function $f$ on $S\times S$, find the upper bound for the number of solutions of the equation $f(p,p') = f(q,q')\neq 0, (p,p',q,q')\in S\times S\times S\times S$; e.g., $f$ is the Euclidean distance in the plane, sphere, or a sheet of the two-sheeted hyperboloid. Our ultimate tool is the Guth--Katz incidence theorem for lines in $\mathbb{RP}^3$, but we focus on how the original problem in two dimensions gets reduced to its application. The corresponding procedure was initiated by Elekes and Sharir, based on symmetry considerations. The point we make here is that symmetry considerations, not necessarily straightforward and potentially requiring group representation machinery can be bypassed or made implicit. The classical Plücker--Klein formalism for line geometry enables one to directly interpret a solution to the above relation as an intersection of two lines in $\mathbb{RP}^3$. This, e.g., allows for a very brief argument as to how the Euclidean plane distance argument extends to the spherical and hyperbolic distances. The space of lines in the projective three-space, the Klein quadric $\mathcal K$, is four dimensional. Thus, we start out with an injective map $\mathfrak F:\,S\times S\to\mathcal K$, that is from a pair of points $(p,q)$ to a line $l_{pq}$ and seek a corresponding combinatorial problem in the above form in two dimensions, which can be solved by applying the Guth--Katz theorem to the set of lines $\{l_{pq}\}$ in $\mathbb{RP}^3$. We identify a few arguably new such problems, and hence applications of the Guth--Katz theorem and make generalizations of the existing ones. It is the dimension of the direct approach in question that is the main purpose of this paper.