  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 GL_{n^2} orbit closure of the determinant, showing
it is an irreducible component of the variety of hypersurfaces of degree $n$ in
C^{n^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.
