Distinguishing secant from cactus varieties

TL;DR

This paper develops new methods to distinguish secant varieties from cactus varieties in Veronese embeddings, providing criteria and algorithms to identify when a point belongs to the secant variety versus the larger cactus variety.

Contribution

It introduces the first techniques to differentiate secant from cactus varieties in Veronese varieties, including explicit descriptions and an algorithm for membership testing.

Findings

The component of the cactus variety not in the secant variety consists of polynomials divisible by a (d-3)-rd power of a linear form.

An algorithm is provided for deciding secant membership in cactus varieties for d ≥ 6 and n ≥ 6.

Results extend to Grassmann cactus varieties and offer partial answers for other cactus varieties.

Abstract

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.