We introduce the use of liaison addition to the study of hyperplane arrangements. For an arrangement, $\mathcal A$, of hyperplanes in $\mathbb P^n$, $\mathcal A$ is free if $R/J$ is Cohen-Macaulay, where $J$ is the Jacobian ideal of $\mathcal A$. Terao's conjecture says that freeness of $\mathcal A$ is determined by the combinatorics of the intersection lattice of $\mathcal A$. We study the Cohen-Macaulayness of three other ideals, all unmixed, that are closely related to $\mathcal A$. Let $\overline J = \mathfrak q_1 \cap \dots \cap \mathfrak q_s$ be the intersection of height two primary components of $J$ and $\sqrt{J} = \mathfrak p_1 \cap \dots \cap \mathfrak p_s$ be the radical of $J$. Our third ideal is $\mathfrak p_1^{b_1} \cap \dots \cap \mathfrak p_s^{b_s}$ for suitable $b_1,\dots, b_s$. With a mild hypothesis we use liaison addition to show that all of these ideals are Cohen-Macaulay. When our hypothesis does not hold, we show that these ideals are not necessarily Cohen-Macaulay, and that Cohen-Macaulayness of any of these ideals does not imply Cohen-Macaulayness of any of the others. While we do not study the freeness of $\mathcal A$, we show by example that the Betti diagrams can vary even for arrangements with the same combinatorics. We then study the situation when the hypothesis does not hold. For equidimensional curves in $\mathbb P^3$, the Hartshorne-Rao module from liaison theory measures the failure of an ideal to be Cohen-Macaulay, degree by degree, and also determines the even liaison class of such a curve. We show that for any positive integer $r$ there is an arrangement $\mathcal A$ for which $R/\overline J$ fails to be Cohen-Macaulay in only one degree, and this failure is by $r$; we also give an analogous result for $\sqrt{J}$. We draw consequences for the corresponding even liaison class of the curve defined by $\overline J$ or by $\sqrt{J}$.

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$\mathbb P^n$ 中超平面排列的奇异轨迹支持的方案

我们在超平面排列的研究中引入了联络加法的使用。对于 $\mathbb P^n$ 中超平面的排列 $\mathcal A$，如果 $R/J$ 是 Cohen-Macaulay，则 $\mathcal A$ 是自由的，其中 $J$ 是 $\ 的雅可比理想数学 A$。Terao 的猜想说 $\mathcal A$ 的自由度是由 $\mathcal A$ 的交点格的组合决定的。我们研究了其他三个与 $\mathcal A$ 密切相关的未混合理想的 Cohen-Macaulayness。令 $\overline J = \mathfrak q_1 \cap \dots \cap \mathfrak q_s$ 是 $J$ 和 $\sqrt{J} 的高度两个主要分量的交集 = \mathfrak p_1 \cap \dots \cap \mathfrak p_s$ 是 $J$ 的部首。我们的第三个理想是 $\mathfrak p_1^{b_1} \cap \dots \cap \mathfrak p_s^{b_s}$ 对于合适的 $b_1,\dots, b_s$。在一个温和的假设下，我们使用联络加法来证明所有这些理想都是 Cohen-Macaulay。当我们的假设不成立时，我们表明这些理想不一定是 Cohen-Macaulay，并且这些理想中的任何一个的 Cohen-Macaulayness 并不意味着其他任何一个的 Cohen-Macaulayness。虽然我们不研究 $\mathcal A$ 的自由度，但我们通过例子表明即使对于具有相同组合的安排，Betti 图也可能有所不同。然后我们研究假设不成立的情况。对于 $\mathbb P^3$ 中的等维曲线，来自联络理论的 Hartshorne-Rao 模块逐次测量理想为 Cohen-Macaulay 的失败，并确定此类曲线的偶联络类。我们证明，对于任何正整数 $r$ 存在一个排列 $\mathcal A$，其中 $R/\overline J$ 仅在一个度上不能是 Cohen-Macaulay，并且这个失败是由 $r$ 引起的；我们也给出了 $\sqrt{J}$ 的类似结果。我们为由 $\overline J$ 或 $\sqrt{J}$ 定义的曲线的相应偶数联络类绘制结果。