关于$\mathbb P^2$中点的线性构形的第一无穷小邻域
On the first infinitesimal neighborhood of a linear configuration of points in $\mathbb P^2$
摘要：
我们考虑以下未决问题。修复一个希尔伯特函数$h$，它出现在$\mathbb P^2$的简化零维子方案中。在所有的子格式$X$中，有Hilbert函数$h$，对于$X$的第一无穷小邻域$Z$，可能的Hilbert函数和分次Betti数是什么（即$X$上支持的双点格式）？是否有最小值（$h^{\min}$）和最大值（$h^{\max}$）这样的函数？编码在$h$中的数值信息转换为{ \it类型向量}，这使我们能够使用希尔伯特函数$h$找到线上点的并集，称为{ \it线性配置}。给出了{ \it all}这类线性构形的第一无穷小邻域的Hilbert函数和分次Betti数相同的充要条件.即使对于那些$h$的Hilbert函数或分级贝蒂数的结果双点计划不是唯一确定的，我们给一个（仅取决于$h$），确实发生。我们证明了$h^{\max}$的存在性，在一般情况下，并讨论了$h^{\min}$。我们的方法包括联络技巧。
英文摘要：
We consider the following open questions. Fix a Hilbert function, $h$, that occurs for a reduced zero-dimensional subscheme of $\mathbb P^2$. Among all subschemes, $X$, with Hilbert function $h$, what are the possible Hilbert functions and graded Betti numbers for the first infinitesimal neighborhood, $Z$, of $X$ (i.e. the double point scheme supported on $X$)? Is there a minimum ($h^{\min}$) and maximum ($h^{\max}$) such function? The numerical information encoded in $h$ translates to a {\it type vector}, which allows us to find unions of points on lines, called {\it linear configurations}, with Hilbert function $h$. We give necessary and sufficient conditions for the Hilbert function and graded Betti numbers of the first infinitesimal neighborhoods of {\it all} such linear configurations to be the same. Even for those $h$ for which the Hilbert functions or graded Betti numbers of the resulting double point schemes are not uniquely determined, we give one (depending only on $h$) that does occur. We prove the existence of $h^{\max}$, in general, and discuss $h^{\min}$. Our methods include liaison techniques.
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