Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

Abstract

We determine set-theoretic defining equations for the variety ${Dual}_{k,d,N}\subset \mathbb{P}(S^d{\mathbb{C}}^N)$ 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.