正割与仙人掌品种的鉴别
Distinguishing secant from cactus varieties
摘要：
仙人掌变种是割线变种的推广。它们是使用有限长度的任意有限格式的线性跨度定义的，而割线品种只使用孤立的约化点。特别是，任何割线品种总是包含在各自的仙人掌品种，而且，除了在少数初始情况下，包含是严格的。众所周知，许多测试正割品种成员资格的自然标准实际上只是测试仙人掌品种的成员资格。在这篇文章中，我们提出了第一个技术来区分实际割线品种的仙人掌品种的情况下，Veronese品种。我们专注于两个初始的情况下，$κ_{14}(ν_d(\mathbb{P}^n))$和$κ_{8,3}(ν_d(\mathbb{P}^n))$，最简单的表现出仙人掌和割线品种之间的差异。我们发现，$d\geq 5$，仙人掌品种$κ_{14}(ν_d(\mathbb{P}^6))$的其他割线品种$σ_{14}(ν_d(\mathbb{P}^6))$的组件由度$d$多项式可分为一个$(d-3)$ -rd权力的线性形式。我们将这种描述推广到任意数量的变量。我们提出了一个算法，用于确定仙人掌品种$κ_{14}(ν_d(\mathbb{P}^n))$中的一个点是否属于正割品种$σ_{14}(ν_d(\mathbb{P}^n))$为$d\geq 6,$ $n \geq 6$。我们得到类似的结果格拉斯曼仙人掌品种$κ_{8,3}(ν_d(\mathbb{P}^n))$。我们的中间结果也给出了部分答案类似的问题，其他仙人掌品种和格拉斯曼仙人掌品种的任何Veronese品种。
英文摘要：
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, $κ_{14}(ν_d(\mathbb{P}^n))$ and $κ_{8,3}(ν_d(\mathbb{P}^n))$, the simplest that exhibit the difference between cactus and secant varieties. We show that for $d\geq 5$, the component of the cactus variety $κ_{14}(ν_d(\mathbb{P}^6))$ other than the secant variety $σ_{14}(ν_d(\mathbb{P}^6))$ 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 $κ_{14}(ν_d(\mathbb{P}^n))$ belongs to the secant variety $σ_{14}(ν_d(\mathbb{P}^n))$ for $d\geq 6,$ $n \geq 6$. We obtain similar results for the Grassmann cactus variety $κ_{8,3}(ν_d(\mathbb{P}^n))$. Our intermediate results give also a partial answer to analogous problems for other cactus varieties and Grassmann cactus varieties to any Veronese variety.
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