标题： Hilbert 8点方案 显示英文标题

标题： Hilbert schemes of 8 points

摘要： 希尔伯特概形 H^d_n 是 A^d 中 n 个点的概形，其中包含一个不可约分支 R^d_n，该分支通常表示 A^d 中 n 个不同的点。我们证明当 n 最多为 8 时，希尔伯特概形 H^d_n 是不可约的当且仅当 n = 8 且 d >= 4。在不可约性的最简单情况下，分支 R^4_8\subset H^4_8 是由一个显式方程定义的，该方程可作为判断给定理想是否为不同点极限的标准。为了理解希尔伯特概形的分支，我们研究了 H_n^d 的闭子概形，这些子概形参数化那些齐次的理想，并具有固定的希尔伯特函数。这些子概形是多分次希尔伯特概形的一个特例，当余长最多为 8 时，我们描述了它们的分支。特别是，我们证明对应于希尔伯特函数 (1,3,2,1) 的概形是最小的不可约例子。 显示英文摘要

摘要： The Hilbert scheme H^d_n of n points in A^d contains an irreducible component R^d_n which generically represents n distinct points in A^d. We show that when n is at most 8, the Hilbert scheme H^d_n is reducible if and only if n = 8 and d >= 4. In the simplest case of reducibility, the component R^4_8 \subset H^4_8 is defined by a single explicit equation which serves as a criterion for deciding whether a given ideal is a limit of distinct points. To understand the components of the Hilbert scheme, we study the closed subschemes of H_n^d which parametrize those ideals which are homogeneous and have a fixed Hilbert function. These subschemes are a special case of multigraded Hilbert schemes, and we describe their components when the colength is at most 8. In particular, we show that the scheme corresponding to the Hilbert function (1,3,2,1) is the minimal reducible example.