Showing papers by "Joseph M. Landsberg published in 2010"

On the Ranks and Border Ranks of Symmetric Tensors

Abstract: Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent.

Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture

Abstract: We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese re-embeddings of arbitrary varieties. Eisenbud's question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that set-theoretic equations of small secant varieties to a high degree Veronese re-embedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However this is false for singular varieties, and we give explicit counter-examples to the EKS conjecture for singular curves. The techniques we use also allow us to prove a gap and uniqueness theorem for symmetric tensor rank. We put Eisenbud's question in a more general context about the behaviour of border rank under specialisation to a linear subspace, and provide an overview of conjectures coming from signal processing and complexity theory in this context.

Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

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 $n$ 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.

Equations for secant varieties to Veronese varieties

Abstract: We define new classes of modules of equations for secant varieties of Veronese varieties using representation theory and geometry. We also revisit some old modules of equations (catalecticant minors) to determine when they are sufficient to give scheme-theoretic defining equations.

Lines and osculating lines of hypersurfaces

Abstract: We define systems of partial differential equations that govern the infinitesimal variation of lines, and osculating lines, through a point of a hypersurface in projective space. The work answers questions posed by J.-M. Hwang.

On the Debarre-de Jong and Beheshti-Starr conjectures on hypersurfaces with too many lines

Abstract: Conjecture 1.1. Let Xn−1 ⊂ P n be a hypersurface of degree d ≥ n and let F(X) ⊂ G(2, n+ 1) denote the Fano scheme of lines on X. Let B ⊂ F(X) be an irreducible component of dimension at least n − 2. Let IB := {(x,E) | x ∈ X, E ∈ B, x ∈ PE}, and let π and ρ denote (respectively) the projections to X and B. Let XB = π(IB) ⊆ X and let Cx = πρ−1ρπ−1(x). Then, for all x ∈XB , Cx ∩Xsing = ∅. If we take hyperplane sections in the case d = n, then Conjecture 1.1 would imply the following, which was conjectured independently by Debarre and de Jong.

Equations for secant varieties via vector bundles

Abstract: We introduce vector bundle techniques for finding equations of secant varieties. A test is established that determines when a secant variety is an irreducible component of the zero set of the equations found. We also prove an induction theorem for varieties that are not weakly defective, that allows one to conclude that the zero set of the equations found for s_{r-1}(X) have s_{r-1}(X) as an irreducible component when s_r(X) is an irreducible component of the equations found for it. The techniques are illustrated with examples of homogeneous varieties. We give an algorithm to decompose a general ternary quintic as the sum of seven fifth powers.