On incidences of lines in regular complexes

Abstract

A regular linear line complex is a three-parameter set of lines in space, whose Plücker vectors lie in a hyperplane, which is not tangent to the Klein quadric. Our main result is a bound O(n1/2m3/4+m+n) for the number of incidences between n lines in a complex and m points in F3, where F is a field, and n≤char(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 F3. This implied the new bound O(n3/2) for the number of realisations of the distance, which is a square, for F, where −1 is not a square in the F-analogue of the Erdős single distance problem in R3. Our incidence bound yields, under a natural constraint, a weaker bound O(n1.6), which holds for any distance, including zero, over any F.