Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

Abstract

We determine set-theoretic defining equations for the variety \mathit{Dual}_{k,d,N} \subset \mathbb{P} (S^d\mathbb{C}^N) of hypersurfaces of degree d in \mathbb{C}^N that have dual variety of dimension at most k . We apply these equations to the Mulmuley–Sohoni variety \overline{\mathrm{GL}_{n^2}\cdot [\det_n]} \subset \mathbb{P} (S^n\mathbb{C}^{n^2}) , showing it is an irreducible component of the variety of hypersurfaces of degree n in \mathbb{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 determinantal border-complexity of the permanent.