  We show that the number of incidences between $m$ distinct points and $n$
distinct lines in ${\mathbb R}^4$ is $O\left(2^{c\sqrt{\log m}}
(m^{2/5}n^{4/5}+m) + m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + n\right)$,
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 this factor, the bound is tight in the
worst case.
