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

Abstract

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.

$3 \times 3$ minors of catalecticants

“…One of them is implicit in Geramita's question Q4 from [Ger99]: Conjecture 1.3 has been proven to be false for singular X [BGL10], but there are no known smooth counterexamples. The case X = P r is a sufficiently interesting special case.…”

Ranks of Linear Subspaces With Respect to an Embedded Variety

“…Fix X ⊂ P r and a positive integer a ≤ ρ ′ (X). Then σ a (X) is the union of all linear spaces Z , where Z ⊂ X is a smoothable zerodimensional scheme of degree a ( [6], Proposition 11, [8], [7]). …”

On the third secant variety

“…We prove an analogous result for σ 3 (Seg(PA × PB × PC)): 3 (Seg(PA × PB × PC)). Moreover, if dim A, dim B, dim C ≥ 3, and p is a general point in the set of the points contained in some P(C 2 ⊗ C 3 ⊗ C 3 ), then p is a non-singular point of σ 3 (Seg(PA×PB ×PC)), and similarly for permuted statements.…”

“…Exceptional limit points turn out to be important-an exceptional limit in σ 5 (Seg(PA × PB × PC)) is used in Bini's approximate algorithm to multiply 2 × 2 matrices with an entry zero, and an exceptional limit in σ 7 (Seg(PA × PB × PC)) is used in Schönhage's approximate algorithm to multiply 3 × 3 matrices using 21 multiplications, see [3,Thm 1.15].…”

“…Normal forms for σ 3 The generalized cominuscule varieties with σ 2 (X) = PV are Seg(P 1 × P n ), Seg(P 1 × P 1 × P 1 ), quadric hypersurfaces Q, the Veronese varieties v 2 (P 1 ), v 3 (P 1 ), the Grassmannians G(2, 5) and G (3,6), the spinor varieties S 5 and S 6 , the Lagrangian Grassmannian G Lag (3,6), Seg(P 1 × Q), and the Freudenthal variety E 7 /P 7 .…”