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

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.Comment: 21 pages; presentation improved as suggested by the referees; To appear in Journal of London Mathematical Societ

A criterion for detecting the identifiability of symmetric tensors of size three

We prove a criterion for the identifiability of symmetric tensors $P$ of type $3\times ...\times 3$, $d$ times, whose rank $k$ is bounded by $(d^2+2d)/8$. The criterion is based on the study of the Hilbert function of a set of points $P_1,..., P_k$ which computes the rank of the tensor $P$

Minimal decomposition of binary forms with respect to tangential projections

Let $C\subset \mathbb{P}^n$ be a rational normal curve and let $\ell_O:\mathbb{P}^{n+1}\dashrightarrow \mathbb{P}^n$ be any tangential projection form a point $O\in T_AC$ where $A\in C$. Hence $X:= \ell_O(C)\subset \mathbb{P}^n$ is a linearly normal cuspidal curve with degree $n+1$. For any $P = \ell_O(B)$, $B\in \mathbb{P}^{n+1}$, the $X$-rank $r_X(P)$ of $P$ is the minimal cardinality of a set $S\subset X$ whose linear span contains $P$. Here we describe $r_X(P)$ in terms of the schemes computing the $C$-rank or the border $C$-rank of $B$.Comment: 7 page

Decomposition of homogeneous polynomials with low rank

Let $F$ be a homogeneous polynomial of degree $d$ in $m+1$ variables defined over an algebraically closed field of characteristic zero and suppose that $F$ belongs to the $s$-th secant varieties of the standard Veronese variety $X_{m,d}\subset \mathbb{P}^{{m+d\choose d}-1}$ but that its minimal decomposition as a sum of $d$-th powers of linear forms $M_1, ..., M_r$ is $F=M_1^d+... + M_r^d$ with $r>s$. We show that if $s+r\leq 2d+1$ then such a decomposition of $F$ can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of $F$ if the rank is at most $d$ and a mild condition is satisfied.Comment: final version. Math. Z. (to appear

Symmetric Tensor Rank and Scheme Rank: An Upper Bound in terms of Secant Varieties

Let X⊂ℙr be an integral and nondegenerate variety. Let c be the minimal integer such that ℙr is the c-secant variety of X, that is, the minimal integer c such that for a general O∈ℙr there is S⊂X with #(S)=c and O∈〈S〉, where 〈 〉 is the linear span. Here we prove that for every P∈ℙr there is a zero-dimensional scheme Z⊂X such that P∈〈Z〉 and deg(Z)≤2c; we may take Z as union of points and tangent vectors of Xreg

Sets computing the symmetric tensor rank

Let n_d denote the degree d Veronese embedding of a projective space P^r. For any symmetric tensor P, the 'symmetric tensor rank' sr(P) is the minimal cardinality of a subset A of P^r, such that n_d(A) spans P. Let S(P) be the space of all subsets A of P^r, such that n_d(A) computes sr(P). Here we classify all P in P^n such that sr(P) < 3d/2 and sr(P) is computed by at least two subsets. For such tensors P, we prove that S(P) has no isolated points