论文标题
通过判别安排的广义西尔维斯特和果园问题
The Generalized Sylvester's And Orchard Problems Via Discriminantal arrangement
论文作者
论文摘要
In 1989 Manin and Schechtman defined the discriminantal arrangement $\mathcal{B}(n, k,\mathcal{A})$ associated to a generic arrangement $\mathcal{A}$ of $n$ hyperplanes in a $k$-dimensional space. 1985年，克拉波（Crapo）已经引入了等效的概念，名称是电路的几何形状。 While both those papers were mainly focused on the case in which $\mathcal{B}(n, k,\mathcal{A})$ has a constant combinatorics when $\mathcal{A}$ changes, it turns out that the case in which the combinatorics of $\mathcal{B}(n, k,\mathcal{A})$ changes is quite interesting as it classifies special $ K $维空间中的点配置。在本文中，我们提供了一个示例，说明了众所周知的广义Sylvester和果园问题与$ \ Mathcal {b}（n，k，k，k，\ Mathcal {a}）$之间的联系。特别是我们指出，这种连接如何有助于解决那些旧但仍在开放的问题。
In 1989 Manin and Schechtman defined the discriminantal arrangement $\mathcal{B}(n, k,\mathcal{A})$ associated to a generic arrangement $\mathcal{A}$ of $n$ hyperplanes in a $k$-dimensional space. An equivalent notion was already introduced by Crapo in 1985 with the name of geometry of circuits. While both those papers were mainly focused on the case in which $\mathcal{B}(n, k,\mathcal{A})$ has a constant combinatorics when $\mathcal{A}$ changes, it turns out that the case in which the combinatorics of $\mathcal{B}(n, k,\mathcal{A})$ changes is quite interesting as it classifies special configurations of points in the $k$-dimensional space. In this paper we provide an example of this fact elucidating the connection between the well known generalized Sylvester's and orchard problems and the combinatorics of $\mathcal{B}(n, k,\mathcal{A})$. In particular we point out how this connection could be helpful to address those old but still open problems.
