论文标题
超平面布置的雅各布的几何方面
Geometric aspects of the Jacobian of a hyperplane arrangement
论文作者
论文摘要
$ \ mathbb {p}^2 $中完整的双分图$ k_ {3,3} $的嵌入，引起了行安排和酒吧和关节框架。对于六个顶点的一般放置，线排列的对数模块的分级betti数量是恒定的，但是当Ziegler引起的一个例子表明，当圆锥上的点位于圆锥上时，分级的Betti数字是不同的。 同样，在刚性理论中，平面中$ k_ {3,3} $的通用嵌入是一个无限的刚性僵硬的条形框架框架，但是当点位于圆锥上时，该框架是无限的灵活性。在本文中，我们发展了超平面布置的弱透视图表示理论，以形式化和推广这个例子所暗示的超平面布置与刚性理论之间的惊人联系。特别是，我们试图了解组合学和几何形状的相互作用如何影响与安排相关的代数结构，例如雅各布式理想的饱和。在超平面布置研究中，我们在刚性理论和有趣现象之间建立了示例和构造之间的联系。
An embedding of the complete bipartite graph $K_{3,3}$ in $\mathbb{P}^2$ gives rise to both a line arrangement and a bar-and-joint framework. For a generic placement of the six vertices, the graded Betti numbers of the logarithmic module of derivations of the line arrangement are constant, but an example due to Ziegler shows that the graded Betti numbers are different when the points lie on a conic. Similarly, in rigidity theory a generic embedding of $K_{3,3}$ in the plane is an infinitesimally rigid bar-and-joint framework, but the framework is infinitesimally flexible when the points lie on a conic. In this paper we develop the theory of weak perspective representations of hyperplane arrangements to formalize and generalize the striking connection between hyperplane arrangements and rigidity theory that this example suggests. In particular, we seek to understand how the interplay of combinatorics and geometry influence algebraic structures associated to an arrangement, such as the saturation of the Jacobian ideal of the arrangement. We make connections between examples and constructions from rigidity theory and interesting phenomena in the study of hyperplane arrangements.
