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We determine set-theoretic defining equations for the variety ofhypersurfaces of degree d in an N-dimensional complex vector space that havedual variety of dimension at most k. We apply these equations to theMulmuley-Sohoni variety, the GL_{n^2} orbit closure of the determinant, showingit is an irreducible component of the variety of hypersurfaces of degree $n$ inC^{n^2} with dual of dimension at most 2n-2. We establish additional geometricproperties of the Mulmuley-Sohoni variety and prove a quadratic lower bound forthe determinental border-complexity of the permanent.

Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program