最小链接与Gaeta的一个结果

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摘要翻译：
如果$V$是$N$维射影空间的等维余维$C$子方案，并且$V$通过完全交集$X$与$V'$相连，则如果$X$是包含$V$的最小度余维$C$完全交集，则我们说$V$是{\em最小连}与$V'$。Gaeta证明了如果$V$是余维2的任何算术Cohen-Macaulay(ACM)子格式，则存在一个从$V$开始到完全交的有限最小链序列。1）在余维数2的非ACM情形下，我们证明了对于任意$n\geq3$,存在在偶数联络类中不是极小的子格式的例子，并且不能在任意步骤中最小地连接到一个极小的子格式。2）然而，在射影3-空间中也有非ACM曲线联络类的例子，其中所有元素在有限个步骤中最小连接到一条最小曲线上。3）推广Huneke和Ulrich以前的工作（关于licci情形），我们证明了在非ACM情形下，在任何高余维数下，存在与偶联络类中的极小子格式不存在极小联系的非极小子格式。4)J.Watanabe在许多年前就证明了任何维的余维3分次Gorenstein理想是Licci，在这里我们证明了任何这样的理想在有限个步骤中最小地联系到一个完全交集，并且它允许一个严格地递减CI-biliaisons到一个完全交集的序列，推广了Hartshorne、Sabadini和Schlesinger的工作。
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英文标题：
《Minimal links and a result of Gaeta》
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作者：
Juan Migliore and Uwe Nagel
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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：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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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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英文摘要：
  If $V$ is an equidimensional codimension $c$ subscheme of an $n$-dimensional projective space, and $V$ is linked to $V'$ by a complete intersection $X$, then we say that $V$ is {\em minimally linked} to $V'$ if $X$ is a codimension $c$ complete intersection of smallest degree containing $V$. Gaeta showed that if $V$ is any arithmetically Cohen-Macaulay (ACM) subscheme of codimension two then there is a finite sequence of minimal links beginning with $V$ and arriving at a complete intersection. We extend this work in the following ways: 1) In the codimension 2 non-ACM case, we show that for any $n \geq 3$ there are examples of subschemes that are not minimal in their even liaison class, and cannot be minimally linked in any number of steps to a minimal subscheme. 2) Nevertheless, there are examples of non-ACM liaison classes of curves in projective 3-space where all elements are minimally linked in a finite number of steps to a minimal curve. 3) Extending previous work of the authors with Huneke and Ulrich (about the licci case), we show that also in the non-ACM case in any higher codimension there are non-minimal subschemes that are not minimally linked to a minimal subscheme in the even liaison class. 4) J. Watanabe had shown many years ago that codimension 3 graded Gorenstein ideals of any dimension are licci. Here we show that any such ideal is minimally linked in a finite number of steps to a complete intersection, and that it admits a sequence of strictly decreasing CI-biliaisons down to a complete intersection, extending work of Hartshorne, Sabadini and Schlesinger.
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PDF链接：
https://arxiv.org/pdf/0804.0590

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关键词：ETA mathematics Dimensional Connections Computation liaison minimal complete there 推广