Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

Abstract

We determine set-theoretic defining equations for the variety Dualk;d;N P.SdCN/ of hypersurfaces of degree d in CN that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety GLn2 OEdetn P.SnCn2 /, showing it is an irreducible component of the variety of hypersurfaces of degree n inCn2 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. © Swiss Mathematical Society.