Distinguishing secant from cactus varieties

Cactus varieties are a generalization of secant varieties. They are defined using linear spans of arbitrary finite schemes of bounded length, while secant varieties use only isolated reduced points. In particular, any secant variety is always contained in the respective cactus variety, and, except in a few initial cases, the inclusion is strict. It is known that lots of natural criteria that test membership in secant varieties are actually only tests for membership in cactus varieties. In this article, we propose the first techniques to distinguish actual secant variety from the cactus variety in the case of the Veronese variety. We focus on two initial cases, $\kappa_{14}(\nu_d(\mathbb{P}^n))$ and $\kappa_{8,3}(\nu_d(\mathbb{P}^n))$, the simplest that exhibit the difference between cactus and secant varieties. We show that for $d\geq 5$, the component of the cactus variety $\kappa_{14}(\nu_d(\mathbb{P}^6))$ other than the secant variety $\sigma_{14}(\nu_d(\mathbb{P}^6))$ consists of degree $d$ polynomials divisible by a $(d-3)$-rd power of a linear form. We generalize this description to an arbitrary number of variables. We present an algorithm for deciding whether a point in the cactus variety $\kappa_{14}(\nu_d(\mathbb{P}^n))$ belongs to the secant variety $\sigma_{14}(\nu_d(\mathbb{P}^n))$ for $d\geq 6,$ $n \geq 6$. We obtain similar results for the Grassmann cactus variety $\kappa_{8,3}(\nu_d(\mathbb{P}^n))$. Our intermediate results give also a partial answer to analogous problems for other cactus varieties and Grassmann cactus varieties to any Veronese variety.