Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

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We determine set-theoretic defining equations for the variety of hypersurfaces of degree d in an N-dimensional complex vector space that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety, the GLₙ₂ orbit closure of the determinant, showing it is an irreducible component of the variety of hypersurfaces of degree n in Cⁿ² with dual of dimension at most 2n-2. We establish additional geometric properties of the Mulmuley-Sohoni variety and prove a quadratic lower bound for the determinental border-complexity of the permanent.