摘要翻译:
A^d中n个点的Hilbert格式H^d_n包含一个不可约分量R^d_n,它泛函地表示A^d中的n个不同点。我们证明了当n至多为8时,Hilbert格式H^d_n是可约的当且仅当n=8且d>=4。在最简单的可约性情况下,分量R^4_8\子集H^4_8由一个显式方程定义,该方程作为判定给定理想是否为不同点的极限的判据。为了理解Hilbert格式的组成部分,我们研究了H_n^d的闭子格式,它们将具有固定Hilbert函数的齐次理想参数化。这些子格式是多重Hilbert格式的一个特例,我们描述了它们在colength最多为8时的分量。特别地,我们证明了对应于Hilbert函数(1,3,2,1)的格式是最小可约的例子。
---
英文标题:
《Hilbert schemes of 8 points》
---
作者:
Dustin A. Cartwright, Daniel Erman, Mauricio Velasco, Bianca Viray
---
最新提交年份:
2008
---
分类信息:
一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
--
一级分类:Mathematics 数学
二级分类:Commutative Algebra 交换代数
分类描述:Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环,模,理想,同调代数,计算方面,不变理论,与代数几何和组合学的联系
--
---
英文摘要:
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.
---
PDF链接:
https://arxiv.org/pdf/0803.0341