Cactus scheme, catalecticant minors, and scheme theoretic equations

The $r$-th cactus variety of a subvariety $X$ in a projective space generalizes the $r$-th secant variety of $X$ and it is defined using linear spans of finite subschemes of $X$ of degree $r$. One of its original purposes was to study the vanishing sets of catalecticant minors. In this article, we equip the cactus variety with a scheme structure, via ``relative linear spans'' of families of finite schemes over a potentially non-reduced base. In this way, we are able to study the vanishing scheme of the catalecticant minors. For a sufficiently high degree Veronese variety, we show that $r$-th cactus scheme and the zero scheme of appropriate catalecticant minors agree on a dense open subset which is the complement of the $(r-1)$-th cactus variety (or scheme). This article is the first part of a series. In the follow-up, as an application, we can describe the singular locus of (in particular) secant varieties to high degree Veronese varieties in terms of singularities of the Hilbert scheme. We will also generalize the result to high degree Veronese reembeddings of other varieties and schemes.