Incidences Between Points and Lines in $${\mathbb {R}}^4$$

We show that the number of incidences between m distinct points and n distinct lines in \({\mathbb {R}}^4\) is \(O(2^{c\sqrt{\log m}} (m^{2/5}n^{4/5}+m) + m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + n)\), for a suitable absolute constant c, provided that no 2-plane contains more than s input lines, and no hyperplane or quadric contains more than q lines. The bound holds without the factor \(2^{c\sqrt{\log m}}\) when \(m \le n^{6/7}\) or \(m \ge n^{5/3}\). Except for the factor \(2^{c\sqrt{\log m}}\), the bound is tight in the worst case.