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

TL;DR

This paper investigates the equations defining secant varieties of Veronese embeddings, proving results for smooth varieties, providing counterexamples for singular ones, and exploring implications for tensor rank and related conjectures.

Contribution

It establishes conditions under which equations of secant varieties are determined by ambient equations, and introduces counterexamples for singular cases, advancing understanding of tensor rank and algebraic geometry.

Findings

Set-theoretic equations for smooth secant varieties are determined by ambient equations and linear equations.

Counterexamples show the EKS conjecture fails for singular curves.

A gap and uniqueness theorem for symmetric tensor rank is proved.

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.