Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

We determine set-theoretic defining equations for the variety Dualk,d,N ⊂ P(S d C N ) of hypersurfaces of degree d in C that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety GLn2 · [detn] ⊂ P(S n C n2), showing it is an irreducible component of the variety of hypersurfaces of degree n in C 2 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.