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” (Bertone et al. in Ann Mat Pura Appl (4) 196(1):19–41, 2017) to give a lower bound on the number of irreducible components of the Hilbert scheme.

在给定的具有固定希尔伯特多项式的多项式环中，我们给出强稳定理想之间的“组合邻近”概念。我们证明了该概念保证了希尔伯特方案中相应点的“几何接近度”。我们定义一个图，其顶点对应于高度稳定的理想，并且其边沿对应于成对的相邻理想。每个项的顺序都会引起图形边缘的方向。该有向图描述了在Gröbner退化下希尔伯特方案的点相对于给定的项阶的行为。然后，我们介绍一种多面体风扇，我们称其为希尔伯特方案的格罗布纳风扇。每个最大尺寸的圆锥体对应于由项顺序引起的不同的有向图。该风扇对希尔伯特方案的几个属性进行编码。我们使用这些工具来展示希尔伯特方案的连通性的新证明。最后，我们改进论文“双重通用初始理想和希尔伯特方案”（Bertone等人，Ann Mat Pura Appl（4）196（1）：19–41，2017）中介绍的技术，以给出下界Hilbert方案的不可约成分的数量。