Algebraic geometry and incidence problems
-- Jordan Ellenberg, September 25, 2015

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Let S be a subset of F<sub>q</sub><sup>n</sup> with the "k-plane Furstenberg property":
for every k-plane V, there is a k-plane W parallel to V which
intersects S in at least q<sup>c</sup> points. We prove that such a set has size
at least a constant multiple of q<sup>cn/k</sup>.  This is a generalization of
Kakeya (the case k = c = 1) but doesn't seem resolvable by Dvir's
approach.  The novelty is the method; we prove that the theorem holds,
not only for subsets of F<sub>q</sub><sup>n</sup>, but (suitably geometrized) for
arbitrary 0-dimensional subschemes of A<sup>n</sup> over an arbitrary field, and
reduce the problem by Grobner methods to a simpler one about
G<sub>m</sub>-invariant non-reduced subschemes supported at a point.  (Joint
work with Daniel Erman.)