Department of Mathematics
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Let F be a homogeneous degree d polynomial in n variables, and suppose we have linear forms L_1,...,L_s such that

Then the collection {L_i} (which can be seen geometrically as a set of s hyperplanes) is called a polar s-hedron of F. Now we consider the set of all s-hedra for a fixed F; this set carries the structure of an algebraic variety. The idea is prima facie artificial, but leads to some beautiful geometry.

I will describe some classical examples of such varieties due to Steiner and Clebsch, as well as some recent enumerative results.