  We show that $m$ points and $n$ smooth algebraic surfaces of bounded degree
in $\mathbb{R}^3$ satisfying suitable nondegeneracy conditions can have at most
$O(m^{\frac{2k}{3k-1}}n^{\frac{3k-3}{3k-1}}+m+n)$ incidences, provided that any
collection of $k$ points have at most O(1) surfaces passing through all of
them, for some $k\geq 3$. In the case where the surfaces are spheres and no
three spheres meet in a common circle, this implies there are $O((mn)^{3/4} + m
+n)$ point-sphere incidences. This is a slight improvement over the previous
bound of $O((mn)^{3/4} \beta(m,n)+ m +n)$ for $\beta(m,n)$ an (explicit) very
slowly growing function. We obtain this bound by using the discrete polynomial
ham sandwich theorem to cut $\mathbb{R}^3$ into open cells adapted to the set
of points, and within each cell of the decomposition we apply a Turan-type
theorem to obtain crude control on the number of point-surface incidences. We
then perform a second polynomial ham sandwich decomposition on the irreducible
components of the variety defined by the first decomposition. As an
application, we obtain a new bound on the maximum number of unit distances
amongst $m$ points in $\mathbb{R}^3$.
