摘要翻译：
设$R=K[x_1,...,x_r]$是无穷域$K$上$R$变量的多项式环，设$M$是$R$的极大理想。这里，a\emph{level algebra}将是$R$的分次Artinian商$a$具有socle$soc(a)=0:m$在一个度数$J$中。Hilbert函数$H(A)=(h_0,h_1,...,h_j)$给出了$A$的每一个度-$I$分级片的维数$H_i=\dim_k a_i$对于$0\lei\lej$。$A$的嵌入维度是$H_1$，$A$的\emph{type}是$\dim_k\soc(A)$，这里是$H_J$。具有Hilbert函数$H$的层次代数商族$\levalg(H)$构成分次代数族的一个开子方案，或者通过Macaulay对偶构成Grassmannian的一个开子方案。我们证明了对于每一个Hilbert函数$H=H_1=(1,3,4,4)$和$H=H_2=(1,3,6,8,9,3)$,Hilbert函数$H$的族$levalg(H)$参数化级Artinian代数都有几个不可约分量。我们还表明，这些例子每个提升到点。然而，在第一个例子中，Artinian代数的一个不可约Betti层在提升到点时变得可约。这是我们在嵌入维度3中获得的$\levalg(H)$的多个组件的第一个示例。我们还证明了第二个例子是第三类Hilbert函数$H(c)$的无穷多个例子中的第一个例子，其中LevAlg(H)的分量数也是任意大的。第一种可能出现多组件现象的情况（即最低嵌入维数然后最低类型）是维数三和类型二。作者和J.-O.都获得了第一种情况的例子。克莱普。
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英文标题：
《Reducible family of height three level algebras》
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作者：
Mats Boij and Anthony Iarrobino
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最新提交年份：
2008
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分类信息：

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
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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
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英文摘要：
  Let $R=k[x_1,..., x_r]$ be the polynomial ring in $r$ variables over an infinite field $k$, and let $M$ be the maximal ideal of $R$. Here a \emph{level algebra} will be a graded Artinian quotient $A$ of $R$ having socle $Soc(A)=0:M$ in a single degree $j$. The Hilbert function $H(A)=(h_0,h_1,... ,h_j)$ gives the dimension $h_i=\dim_k A_i$ of each degree-$i$ graded piece of $A$ for $0\le i\le j$. The embedding dimension of $A$ is $h_1$, and the \emph{type} of $A$ is $\dim_k \Soc (A)$, here $h_j$. The family $\Levalg (H)$ of level algebra quotients of $R$ having Hilbert function $H$ forms an open subscheme of the family of graded algebras or, via Macaulay duality, of a Grassmannian.   We show that for each of the Hilbert functions $H=H_1=(1,3,4,4)$ and $H=H_2=(1,3,6,8,9,3)$ the family $LevAlg (H)$ parametrizing level Artinian algebras of Hilbert function $H$ has several irreducible components. We show also that these examples each lift to points. However, in the first example, an irreducible Betti stratum for Artinian algebras becomes reducible when lifted to points. These were the first examples we obtained of multiple components for $\Levalg(H)$ in embedding dimension three.   We also show that the second example is the first in an infinite sequence of examples of type three Hilbert functions $H(c)$ in which also the number of components of LevAlg(H) gets arbitrarily large.   The first case where the phenomenon of multiple components can occur (i.e. the lowest embedding dimension and then the lowest type) is that of dimension three and type two. Examples of this first case have been obtained by the authors and also by J.-O. Kleppe.
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PDF链接：
https://arxiv.org/pdf/0707.2148