  We establish the sharp growth order, up to epsilon losses, of the $L^2$-norm
of the maximal directional averaging operator along a finite subset $V$ of a
polynomial variety of arbitrary dimension $m$, in terms of cardinality. This is
an extension of the works by C\'ordoba, for one-dimensional manifolds, Katz for
the circle in two dimensions, and Demeter for the 2-sphere. For the case of
directions on the two-dimensional sphere we improve by a factor of $\sqrt{\log
N}$ on the best known bound, due to Demeter, and we obtain a sharp estimate for
our model operator. Our results imply new $L^2$-estimates for Kakeya-type
maximal functions with tubes pointing along polynomial directions. Our proof
technique is novel and in particular incorporates an iterated scheme of
polynomial partitioning on varieties adapted to directional operators, in the
vein of Guth, Guth-Katz, and Zahl.
