论文标题
在非常通用的判别安排上
On Very Generic Discriminantal Arrangements
论文作者
论文摘要
在本文中，我们证明了两个主要结果。首先，我们表明，任何六线排列，由三对相互垂直线组成，都不会从C. a. athanasiadis \ cite {MR1720104}的意义上产生“非常通用或足够通用的判别排列”。我们给出了两个结果的两个证明。第二个结果如下。 （一个区域）一个非常通用的判别排列的凸锥的编态度 - 一个边界面尚未被表征，即使已知非常通用的判别排列的相交晶格也不知道。其次，我们表明，非常通用的超平面排列的单纯单元数$ \ MATHCAL {等于在相关的非常通用的歧视安排中，构成$（C_1，C_2，\ ldots，c_n）$的$ \ mathbb {r}^n $的$ \ mathbb {r}^n $的边界超平面数。也就是说，价格为$ 1 \ leq i_1 <i_2 <\ ldots <i_m <i_ {m+1} \ leq n $，如果$δ^m h_ {i_1} h_ {i_2} \ ldots h_ {i_2} \ ldots H_ $ \ MATHCAL {h}^m_n $，那么它不必引起convex锥$ c $ cone $ c $ contemension-cone $ c $（C_1，c_1，c_2，\ ldots，c_n）$，在关联的非常通用的非常通用的歧视安排中。我们终于在附录部分提到了一个有趣的开放式备注。 在附录部分中，我们给出了一个自我包含的博览会，并在组合中描述了“非常通用或足够通用的”判别安排（Zariski开放且密集）类的相交晶格。结果，我们将晶格元素的几何描述作为超平面布置的一组，赋予相同的“非常通用或足够通用的”判别排列。
In this article we prove two main results. Firstly, we show that any six-line arrangement, consisting of three pairs of mutually perpendicular lines, does not give rise to a "very generic or sufficiently general" discriminantal arrangement in the sense of C. A. Athanasiadis \cite{MR1720104}. We give two proofs of the first result. The second result is as follows. The codimension-one boundary faces of (a region) a convex cone of a very generic discriminantal arrangement has not been characterized and is not known even though the intersection lattice of a very generic discriminantal arrangement is known. So secondly, we show that the number of simplex cells of the very generic hyperplane arrangement $\mathcal{H}^m_n=\{H_i:\underset{j=1}{\overset{m}{\sum}}a_{ij}x_j=c_i,1\leq i\leq n\}$ may not be not precisely equal to the number of codimension-one boundary hyperplanes of $\mathbb{R}^n$ of the convex cone $C$ containing $(c_1,c_2,\ldots,c_n)$ in the associated very generic discriminantal arrangement. That is, for $1\leq i_1<i_2<\ldots<i_m<i_{m+1}\leq n$, if $Δ^m H_{i_1}H_{i_2}\ldots H_{i_m}H_{i_{m+1}}$ is a simplex cell of the hyperplane arrangement $\mathcal{H}^m_n$ then it need not give rise to a codimension-one boundary hyperplane of the convex cone $C$ containing $(c_1,c_2,\ldots,c_n)$ in the associated very generic discriminantal arrangement. We finally mention an interesting open-ended remark before the appendix section. In the appendix section we give a self contained exposition and describe combinatorially the intersection lattice of a (Zariski open and dense) class of "very generic or sufficiently general" discriminantal arrangements. As a consequence, we give a geometric description of the lattice elements as sets of concurrencies of the hyperplane arrangements which give the same "very generic or sufficiently general" discriminantal arrangement.
