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.

21 pages; presentation improved as suggested by the referees; To appear in Journal of London Mathematical Society

References in corpus (9)

• Computing symmetric rank for symmetric tensors
• On degeneration of tensors and algebras
• Ranks of tensors and a generalization of secant varieties
• 3x3 Minors of Catalecticants
• The Solution to Waring's Problem for Monomials
• A generalization of the Clifford index and determinantal equations for curves and their secant varieties
• On the third secant variety
• Irreducibility of the Gorenstein locus of the punctual Hilbert scheme of degree 10
• Inverse Systems of Zero-dimensional Schemes in P^n

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• Partially symmetric variants of Comon's problem via simultaneous rank
• Real identifiability vs complex identifiability
• Deformations of zero-dimensional schemes and applications
• A counterexample to Strassen's direct sum conjecture
• On the geometry of border rank algorithms for matrix multiplication and other tensors with symmetry
• Unique decomposition for a polynomial of low rank
• Lower bounds on the rank and symmetric rank of real tensors
• Geometric conditions for strict submultiplicativity of rank and border rank
• Brill's equations as a GL(V)-module
• Bounds on the tensor rank
• On a geometric method for the identifiability of forms
• Finite schemes and secant varieties over arbitrary characteristic
• Almost maximal growth of the Hilbert function
• Dependent subsets of embedded projective varieties
• Entanglement in the symmetric subspace: mapping multipartite to bipartite states
• On Comon's and Strassen's conjectures
• Sets computing the symmetric tensor rank
• A uniqueness result on the decompositions of a bi-homogeneous polynomial
• Remarks on the symmetric rank of symmetric tensors
• On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank
• On the stratification by $X$-ranks of a linearly normal elliptic curve $X\subset {\mathbb{P}}^n$
• The stratification by rank for homogeneous polynomials with border rank 5 which essentially depend on 5 variables