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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 counterexamples to the EKS conjecture for singular curves. The techniques that 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 specialization to a linear subspace and provide an overview of conjectures coming from signal processing and complexity theory in this context.

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