On incidences of lines in regular complexes

Abstract: A regular linear line complex is a three-parameter set of lines in space, whose Pl\"ucker vectors lie in a hyperplane, which is not tangent to the Klein quadric. Our main result is a bound $O(n^{1/2}m^{3/4} + m+n)$ for the number of incidences between $n$ lines in a complex and $m$ points in $\mathbb F^3$, where $\mathbb F$ is a field, and $n\leq char(\mathbb F)^{4/3}$ in positive characteristic. Zahl has recently observed that bichromatic pairwise incidences of lines coming from two distinct line complexes account for the nonzero single distance problem for a set of $n$ points in $\mathbb F^3$. This implied the new bound $O(n^{3/2})$ for the number of realisations of the distance, which is a square, for $\mathbb F$, where $-1$ is not a square in the $\mathbb F$-analogue of the Erd\H os single distance problem in $\mathbb R^3$. Our incidence bound yields, under a natural constraint, a weaker bound $O(n^{1.6})$, which holds for any distance, including zero, over any $\mathbb F$.