  We give a lower bound for the size of a subset of $\mathbb F_q^n$ containing
a rich k-plane in every direction, a k-plane Furstenberg set. The chief novelty
of our method is that we use arguments on non-reduced subschemes and flat
families to derive combinatorial facts about incidences between points and
k-planes in space.

Algebraic Geometry;Classical Analysis and ODEs;Combinatorics