数值粗化
Numerical Macaulification
摘要：
1983年（或更早）的一个未发表的例子，由Joe Harris给出了两条具有相同希尔伯特函数的光滑空间曲线，但其中一条曲线是算术Cohen-Macaulay（ACM），另一条不是。从任意多个变量的任意齐次理想出发，给出了两种构造方法，每一种构造方法在有限步内构造出一个余维为2的ACM子概型的Hilbert函数理想.我们把与这种理想相联系的子概型称为“数值ACM”。“我们研究了这两种构造之间的联系，特别是表明它们产生具有相同希尔伯特函数的理想。我们称从任一构造得到的理想为原始理想的“数值Macaulification”。专门的情况下，理想是unmixed的余维2，我们表明：（a）每一个甚至联络类，$\mathcal L$，包含数值ACM子计划，（b）的子集，$\mathcal M$，数值ACM子计划在$\mathcal L$有，本身，一个Lazarssyn-Rao结构，和（c）数值Macaulification的最小元素的$\mathcal L$是一个极小元的$\mathcal M$。最后，如果我们进一步限制到$\mathbb P^3$中的曲线，我们证明了对于所有的$n \geq 1$，具有Hartshorne-Rao模的曲线的偶联络类集中在一个度上，并且具有维数$n$，包含光滑的，数值上的ACM曲线。第一个（也是最小的）例子是哈里斯。我们的结果的一个后果是，知识的Hilbert函数的积分曲线单独是不足以决定它是否包含零维算术Gorenstein子计划的任意大的程度。
英文摘要：
An unpublished example due to Joe Harris from 1983 (or earlier) gave two smooth space curves with the same Hilbert function, but one of the curves was arithmetically Cohen-Macaulay (ACM) and the other was not. Starting with an arbitrary homogeneous ideal in any number of variables, we give two constructions, each of which produces, in a finite number of steps, an ideal with the Hilbert function of a codimension two ACM subscheme. We call the subscheme associated to such an ideal "numerically ACM." We study the connections between these two constructions, and in particular show that they produce ideals with the same Hilbert function. We call the resulting ideal from either construction a "numerical Macaulification" of the original ideal. Specializing to the case where the ideals are unmixed of codimension two, we show that (a) every even liaison class, $\mathcal L$, contains numerically ACM subschemes, (b) the subset, $\mathcal M$, of numerically ACM subschemes in $\mathcal L$ has, by itself, a Lazarsfeld-Rao structure, and (c) the numerical Macaulification of a minimal element of $\mathcal L$ is a minimal element of $\mathcal M$. Finally, if we further restrict to curves in $\mathbb P^3$, we show that the even liaison class of curves with Hartshorne-Rao module concentrated in one degree and having dimension $n$ contains smooth, numerically ACM curves, for all $n \geq 1$. The first (and smallest) such example is that of Harris. A consequence of our results is that the knowledge of the Hilbert function of an integral curve alone is not enough to decide whether it contains zero-dimensional arithmetically Gorenstein subschemes of arbitrarily large degree.
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