Cactus scheme, catalecticant minors, and scheme theoretic equations

TL;DR

This paper introduces a scheme-theoretic framework for cactus varieties, linking them with catalecticant minors and enabling detailed geometric and algebraic analysis of secant varieties.

Contribution

It develops a scheme structure for cactus varieties using relative linear spans, connecting them with catalecticant minors and advancing the understanding of their geometric properties.

Findings

Cactus scheme and catalecticant minors agree on a dense open subset for high-degree Veronese varieties.

The scheme structure allows studying the vanishing scheme of catalecticant minors.

Foundation laid for analyzing singularities of secant varieties via Hilbert schemes.

Abstract

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.