  Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as
points in the four-dimensional Pl\"ucker space of lines in $R^3$, in an
irreducible variety $T$ of constant degree which is \emph{non-degenerate} with
respect to $L$ (see below). We show:
  \medskip \noindent{\bf (1)} If $T$ is two-dimensional, the number of $r$-rich
points (points incident to at least $r$ lines of $L$) is
$O(n^{4/3+\epsilon}/r^2)$, for $r \ge 3$ and for any $\epsilon>0$, and, if at
most $n^{1/3}$ lines of $L$ lie on any common regulus, there are at most
$O(n^{4/3+\epsilon})$ $2$-rich points. For $r$ larger than some sufficiently
large constant, the number of $r$-rich points is also $O(n/r)$.
  As an application, we deduce (with an $\epsilon$-loss in the exponent) the
bound obtained by Pach and de Zeeuw (2107) on the number of distinct distances
determined by $n$ points on an irreducible algebraic curve of constant degree
in the plane that is not a line nor a circle.
  \medskip \noindent{\bf (2)} If $T$ is two-dimensional, the number of
incidences between $L$ and a set of $m$ points in $R^3$ is $O(m+n)$.
  \medskip \noindent{\bf (3)} If $T$ is three-dimensional and nonlinear, the
number of incidences between $L$ and a set of $m$ points in $R^3$ is
$O\left(m^{3/5}n^{3/5} + (m^{11/15}n^{2/5} + m^{1/3}n^{2/3})s^{1/3} + m + n
\right)$, provided that no plane contains more than $s$ of the points. When $s
= O(\min\{n^{3/5}/m^{2/5}, m^{1/2}\})$, the bound becomes
$O(m^{3/5}n^{3/5}+m+n)$.
  As an application, we prove that the number of incidences between $m$ points
and $n$ lines in $R^4$ contained in a quadratic hypersurface (which does not
contain a hyperplane) is $O(m^{3/5}n^{3/5} + m + n)$.
  The proofs use, in addition to various tools from algebraic geometry, recent
bounds on the number of incidences between points and algebraic curves in the
plane.
