Incidences Between Points and Lines in R4

We show that the number of incidences between m distinct points and n distinct lines in R 4 is O ( 2 c 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 log m when m ≤ n 6 / 7 or m ≥ n 5 / 3 . Except for the factor 2 c log m , the bound is tight in the worst case.