Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

Details Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

2010-04-27

arxiv.org/abs/1004.4802v1

Mathematics

Algebraic Geometry

Scientific paper

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 $n$ 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.

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