Hilbert schemes of 8 points
Hilbert schemes of 8 points
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8 点希尔伯特方案
DOI：
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发表时间：
2008
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通讯作者：
B. Viray
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作者：
Dustin A. Cartwright;D. Erman;Mauricio Velasco;B. Viray
a^d中n个点的Hilbert方案H^d_n包含一个不可约的组件r^d_n，该组件在A^d中通常代表n个不同的点。我们表明，当n最多是8时，Hilbert方案H^d_n是可以降低的，并且仅在n = 8和d> = 4。在最简单的降低情况下，组件R^4_8子集H^4_8由单个显式方程式作为确定给定理想是否是不同点的限制的标准。
为了了解希尔伯特方案的组成部分，我们研究了H_n^d的闭合亚化学，该基础是参数为均匀且具有固定希尔伯特功能的理想的参数。这些亚化学是多面化希尔伯特方案的特殊情况，我们描述了当colength最多8个时的成分。特别是，我们表明与希尔伯特函数相对应的方案（1,3,2,1）是最小的还原性。例子。
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.