The Chow Form of a Reciprocal Linear Space
The Chow Form of a Reciprocal Linear Space
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倒数线性空间的 Chow 形式
DOI：
10.1307/mmj/1571731287
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发表时间：
2016
影响因子：
0.9
通讯作者：
C. Vinzant
中科院分区：
文献类型：
--
作者：
Mario Kummer;C. Vinzant
倒易线性空间是线性空间在坐标反演下的图像。这些基本变量描述了超平面排列的解析中心，并作为线性规划中心路径的定义方程的一部分出现。它们的结构由一个基本拟阵控制。这提供了一个大家庭的双曲品种，最近推出的沙莫维奇和Vinnikov。本文给出了倒易线性空间的Chow型的一个确定的行列式表示。一个结果是存在的对称秩一乌尔里希层互惠线性空间。另一种是熵判别式的平方和表示。对于一般线性空间，所得到的行列式公式与完全图的拉普拉斯算子密切相关，并推广到单纯拟阵。这提出了有趣的问题组合的双曲品种和连接与积极的格拉斯曼。
A reciprocal linear space is the image of a linear space under coordinate-wise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal linear spaces. Another is a representation of the entropic discriminant as a sum of squares. For generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph and generalizations to simplicial matroids. This raises interesting questions about the combinatorics of hyperbolic varieties and connections with the positive Grassmannian.