# Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

### Laurent Manivel

Université Grenoble I, Saint-Martin-d'Hères, France### Joseph M. Landsberg

Texas A&M University, College Station, USA### Nicolas Ressayre

Université Claude Bernard Lyon 1, Villeurbanne, France

## 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.

## Cite this article

Laurent Manivel, Joseph M. Landsberg, Nicolas Ressayre, Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program. Comment. Math. Helv. 88 (2013), no. 2, pp. 469–484

DOI 10.4171/CMH/292