A good way of parameterizing zero-dimensional schemes in an affine space [Formula: see text] has been developed in the last 20 years using border basis schemes. Given a multiplicity [Formula: see text], they provide an open covering of the Hilbert scheme [Formula: see text] and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent zero-dimensional [Formula: see text]-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley–Bacharach, and strict Cayley–Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by nontrivial, concrete examples.

中文翻译：

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边界基方案的计算子方案

在过去的 20 年中，使用边界基方案开发了一种在仿射空间中参数化零维方案的好方法 [公式：见正文]。给定多重性 [公式：见正文]，它们提供了希尔伯特方案 [公式：见正文] 的开放覆盖，并且可以用易于计算的二次方程来描述。一个自然的问题是如何确定包含在边界基础方案中的轨迹，并且其有理点表示零维 [公式：参见文本]-共享给定属性的代数。本文的主要重点是对这个普遍问题给出有效的答案。这里考虑的属性是局部 Gorenstein、严格 Gorenstein、严格完全交集、Cayley-Bacharach 和严格 Cayley-Bacharach 属性。我们方法的关键特征是我们通过展示明确的算法来计算它们的定义理想来描述这些基因座。所有结果都通过不平凡的具体例子来说明。