Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program
Algebraic Geometry
2010-04-28 v1 Computational Complexity
Abstract
We determine set-theoretic defining equations for the variety of hypersurfaces of degree d in an N-dimensional complex vector space that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety, the GL_{n^2} orbit closure of the determinant, showing it is an irreducible component of the variety of hypersurfaces of degree in 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 determinental border-complexity of the permanent.
Cite
@article{arxiv.1004.4802,
title = {Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program},
author = {J. M. Landsberg and Laurent Manivel and Nicolas Ressayre},
journal= {arXiv preprint arXiv:1004.4802},
year = {2010}
}