Finite schemes and secant varieties over arbitrary characteristic (Q1679669)

In this very well-written article, the authors establish several results relating finite schemes and secant varieties over arbitrary fields. Some of these results were previously known only over the complex numbers. The paper is in part expository and contains background material on scheme theory, apolarity theory, Castelnuovo-Mumford regularity, Hilbert schemes, and secant varieties. Let \(\mathbb K\) be a field and \(R\) be a finite scheme over \(\mathbb K\). One of the main objectives is to study the \textit{smoothability} of \(R\) both as an abstract scheme and as an embedded scheme in some algebraic variety \(X\). The condition of smoothability can be easily seen over an algebraically closed field: a finite scheme \(R\) is smoothable if and only if it is a flat limit of distinct points. Theorem 1.1 gives the equivalence between the abstract smoothability and the embedded smoothability in some algebraic variety \(X\), whenever \(X\) is smooth. Moreover, smoothability over \(\mathbb K\) is equivalent to smoothability in the algebraic closure of \(\mathbb K\) (Proposition 1.2). Let \(\mathbb K\) be an algebraically closed field. Let \(X\) be an algebraic variety \(\mathbb K\) and let \(r\) be an integer. Condition \((\star)\) holds if every finite \textit{Gorenstein} subscheme over \(\mathbb K\) of \(X\) of degree at most \(r\) is smoothable in \(X\). One of the main results is Theorem 1.7. This relates the scheme theoretic condition above with the possibility of giving \textit{set-theoretic equations} for secants of sufficiently high Veronese embeddings of \(X\), by determinantal equations from vector bundles on \(X\). If condition \((\star)\) does not hold, then those equations are not enough to cut them. Interestingly, the locus of determinantal equations from vector bundles contain more general loci than secants: the \textit{cactus varieties}. This containment is the ultimate reason for the failure of present methods to give good enough lower bounds on tensor ranks.