  We extend (and somewhat simplify) the algebraic proof technique of Guth and
Katz \cite{GK}, to obtain several sharp bounds on the number of incidences
between lines and points in three dimensions. Specifically, we show: (i) The
maximum possible number of incidences between $n$ lines in $\reals^3$ and $m$
of their joints (points incident to at least three non-coplanar lines) is
$\Theta(m^{1/3}n)$ for $m\ge n$, and $\Theta(m^{2/3}n^{2/3}+m+n)$ for $m\le n$.
(ii) In particular, the number of such incidences cannot exceed $O(n^{3/2})$.
(iii) The bound in (i) also holds for incidences between $n$ lines and $m$
arbitrary points (not necessarily joints), provided that no plane contains more
than O(n) points and each point is incident to at least three lines. As a
preliminary step, we give a simpler proof of (an extension of) the bound
$O(n^{3/2})$, established by Guth and Katz, on the number of joints in a set of
$n$ lines in $\reals^3$. We also present some further extensions of these
bounds, and give a proof of Bourgain's conjecture on incidences between points
and lines in 3-space, which constitutes a simpler alternative to the proof of
\cite{GK}.
