  In this paper we establish an improved bound for the number of incidences
between a set $P$ of $m$ points and a set $H$ of $n$ planes in $\mathbb R^3$,
provided that the points lie on a two-dimensional nonlinear irreducible
algebraic variety $V$ of constant degree. Specifically, the bound is $$ O\left(
m^{2/3}n^{2/3} + m^{6/11}n^{9/11}\log^\beta(m^3/n) + m + n + \sum_\ell
|P_\ell|\cdot |H_\ell| \right) , $$ where the constant of proportionality and
the constant exponent $\beta$ depend on the degree of $V$, and where the sum
ranges over all lines $\ell$ that are fully contained in $V$ and contain at
least one point of $P$, so that, for each such $\ell$, $P_\ell = P\cap\ell$ and
$H_\ell$ is the set of the planes of H that contain $\ell$. In addition,
$\sum_\ell |P_\ell| = O(m)$ and $\sum_\ell |H_\ell| = O(n)$.
  This improves, for this special case, the earlier more general bound of
Apfelbaum and Sharir (see also Brass and Knauer as well as Elekes and T\'oth).
  This is a generalization of the incidence bound for points and circles in the
plane (cf. Aronov et al., Aronov and Sharir, Marcus and Tardos), and is based
on a recent result of Sharir and Zahl on the number of cuts that turn a
collection of algebraic curves into pseudo-segments.
  The case where $V$ is a quadric is simpler to analyze, does not require the
result of Sharir and Zahl, and yields the same bound as above, with
$\beta=2/11$.
  We present an interesting application of our results to a problem, studied by
Rudnev, on obtaining a lower bound on the number of distinct cross-ratios
determined by $n$ real points, where our bound leads to a slight improvement in
Rudnev's bound.
