  In this paper we study the mapping properties of the averaging operator over
a variety given by a system of homogeneous equations over a finite field. We
obtain optimal results on the averaging problems over two dimensional varieties
whose elements are common solutions of diagonal homogeneous equations. The
proof is based on a careful study of algebraic and geometric properties of such
varieties. In particular, we show that they are not contained in any hyperplane
and are complete intersections. We also address partial results on averaging
problems over arbitrary dimensional homogeneous varieties which are smooth away
from the origin.
