The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Commentarii Mathematici Helvetici


Full-Text PDF (201 KB) | Metadata | Table of Contents | CMH summary
Volume 88, Issue 2, 2013, pp. 469–484
DOI: 10.4171/CMH/292

Published online: 2013-04-24

Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

Joseph M. Landsberg[1], Laurent Manivel[2] and Nicolas Ressayre[3]

(1) Texas A&M University, College Station, USA
(2) Université Grenoble I, Saint-Martin-d'Hères, France
(3) Université Claude Bernard Lyon 1, Villeurbanne, France

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.

Keywords: Dual variety, geometric complexity theory, determinant, permanent

Landsberg Joseph, Manivel Laurent, Ressayre Nicolas: Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program. Comment. Math. Helv. 88 (2013), 469-484. doi: 10.4171/CMH/292