圆锥的Varchenko-Gel'fand环
The Varchenko-Gel'fand Ring of a Cone
摘要：
对于实向量空间中的超平面排列，其庞加莱多项式的系数有许多解释。一个有趣的是提供了由瓦琴科-凝胶'fand环，这是环的功能，从室的安排，以整数与逐点加法和乘法。Varchenko和Gel'fand给出了一个简单的介绍，这个环，随着过滤和相关的分次环的希尔伯特级数是庞加莱多项式。我们将这些结果推广到由某些超平面的半空间的交点所定义的锥上，并证明了一个关于排列的Varchenko-Gel'fand环的新结果：当排列是超可解的时，该排列的相关分次环是Koszul.
英文摘要：
For a hyperplane arrangement in a real vector space, the coefficients of its Poincaré polynomial have many interpretations. An interesting one is provided by the Varchenko-Gel'fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise addition and multiplication. Varchenko and Gel'fand gave a simple presentation for this ring, along with a filtration and associated graded ring whose Hilbert series is the Poincaré polynomial. We generalize these results to cones defined by intersections of halfspaces of some of the hyperplanes and prove a novel result for the Varchenko-Gel'fand ring of an arrangement: when the arrangement is supersolvable the associated graded ring of the arrangement is Koszul.
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