论文标题
区分切乳属品种
Distinguishing secant from cactus varieties
论文作者
论文摘要
仙人掌品种是割线品种的概括。它们是使用有限长度的任意有限方案的线性跨度来定义的，而Scant品种仅使用隔离的降低点。特别是，任何距离仙人掌品种都始终包含在各自的仙人掌品种中，除了少数初始情况外，包含是严格的。众所周知，许多自然标准将割线品种测试成员资格实际上只是仙人掌品种成员资格的测试。在本文中，我们提出了第一种将实际距离品种与仙人掌品种区分开的技术。我们专注于两种初始情况，$κ_{14}（ν_d（\ Mathbb {p}^n））$和$κ__{8,3}（ν_d（\ Mathbb {p}^n））$，最简单的是仙人掌和叶片之间的差异。我们表明，对于$ d \ geq 5 $，仙人掌品种的组成部分$κ_{14}（ν_d（c \ mathbb {p}^6））$以外的其他$σ_{14}（ν_d（ν_d（c}^6）$（$ divis）$ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ - 线性形式。我们将此描述推广到任意数量的变量。我们提出了一种算法，用于确定仙人掌品种中的一个点$κ_{14}（ν_d（\ Mathbb {p}^n））$属于SCANT品种$σ_{14}（ν_d（ν_d（\ Mathbb {p}^n）$ d for $ d for $ d d f $ d \ egeq 6，$ n $ n $ n $ n $我们获得了Grassmann仙人掌品种$κ_{8,3}（ν_d（\ Mathbb {p}^n））$的类似结果。我们的中级结果还为其他仙人掌品种和Grassmann仙人掌品种的类似问题提供了部分答案。
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.
