论文标题
希尔伯特计划的格鲁布纳（Gröbner）粉丝
The Gröbner fan of the Hilbert scheme
论文作者
论文摘要
我们在具有固定的希尔伯特多项式的给定多项式环中强稳定的理想中给出了“联合近端”的概念。我们表明，这个概念保证了希尔伯特计划中相应点的“几何接近”。我们定义一个图形，其顶点对应于强稳定的理想，其边缘对应于相邻理想对。每个术语顺序诱导图形边缘的方向。该定向图描述了在Gröbner变性下相对于给定期限顺序的Hilbert方案点的行为。 然后，我们介绍了一个多面体风扇，我们称之为希尔伯特计划的Gröbner迷。每个最大尺寸的锥体对应于由期限顺序诱导的不同定向图。该风扇编码了希尔伯特方案的几个属性。我们使用这些工具为希尔伯特计划的连接性提供了新的证明。最后，我们改进了Bertone，Cioffi和Roggero在论文“双生初始理想和希尔伯特方案”中引入的技术，以对希尔伯特方案的不可约组件的数量进行下限。
We give a notion of "combinatorial proximity" among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees "geometric proximity" of the corresponding points in the Hilbert scheme. We define a graph whose vertices correspond to strongly stable ideals and whose edges correspond to pairs of adjacent ideals. Every term order induces an orientation of the edges of the graph. This directed graph describes the behavior of the points of the Hilbert scheme under Gröbner degenerations with respect to the given term order. Then, we introduce a polyhedral fan that we call Gröbner fan of the Hilbert scheme. Each cone of maximal dimension corresponds to a different directed graph induced by a term order. This fan encodes several properties of the Hilbert scheme. We use these tools to present a new proof of the connectedness of the Hilbert scheme. Finally, we improve the technique introduced in the paper "Double-generic initial ideal and Hilbert scheme" by Bertone, Cioffi and Roggero to give a lower bound on the number of irreducible components of the Hilbert scheme.
