JournalscmhVol. 88, No. 2pp. 469–484

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
Hypersurfaces with degenerate duals and the Geometric  Complexity Theory Program cover
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Abstract

We determine set-theoretic defining equations for the variety Dualk,d,NP(SdCN)\mathit{Dual}_{k,d,N} \subset \mathbb{P} (S^d\mathbb{C}^N) of hypersurfaces of degree dd in CN\mathbb{C}^N that have dual variety of dimension at most kk. We apply these equations to the Mulmuley–Sohoni variety GLn2[detn]P(SnCn2)\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 nn in Cn2\mathbb{C}^{n^2} with dual of dimension at most 2n22n-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