  In this note we give a shortened proof of a theorem of Rudnev, which bounds
the number of incidences between points and planes over an arbitrary field.
Rudnev's proof uses a map that goes via the four-dimensional Klein quadric to a
three-dimensional space, where it applies a bound of Guth and Katz on
intersection points of lines. We describe a simple geometric map that directly
sends point-plane incidences to line-line intersections in space, allowing us
to reprove Rudnev's theorem with fewer technicalities.
