  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.
