多重奇点的退化与阿廷代数
Degenerations of multisingularities and Artin algebras
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中文总结 AI 辅助
研究交换、结合、有限维复阿廷代数的退化层次结构，引入基于奇点与局部代数对应的退化概念，通过托姆多项式从对称数据确定层次结构，还证明其扩展了变形理论的代数层次结构，推广到不同维度代数。
中文摘要 AI 辅助
我们研究交换、结合、有限维复阿廷代数的退化层次结构。我们不是在希尔伯特概型中研究退化，而是基于稳定映射芽的奇点与局部代数之间的对应关系，引入一种奇点理论的退化概念。这导致一个自然的偏序集，即稳定层次结构，如果附近的奇点实现了后者，则一个代数退化为另一个代数。我们的第一个主要结果是，在广泛的维度范围内，这个层次结构可以纯粹从对称数据，即代数或奇点的自同构群来确定。关键工具是由卡扎里安建立的某些等变特征类的理论，称为多重奇点的托姆多项式。对这些多项式进行适当的代换完全刻画了层次结构。因此，退化偏序集的计算本质上变得算法化。在我们的第二个主要结果中，我们证明我们的奇点理论层次结构扩展了从变形理论得到的代数层次结构。虽然变形理论要求维度（秩）固定，但我们的层次结构通过比较不同维度的代数来推广这个框架。
英文摘要
We study the degeneration hierarchy of commutative, associative, finite-dimensional complex Artin algebras. Instead of studying degenerations in the Hilbert scheme, we introduce a singularity-theoretic notion of degeneration based on the correspondence between singularities of stable map germs and local algebras. This leads to a natural partially ordered set, the stable hierarchy, in which one algebra degenerates to another if nearby singularities realize the latter. Our first main result is that, in a wide range of dimensions, this hierarchy can be determined purely from symmetry data, namely from the automorphism groups of the algebras or singularities. The key tool is the theory of certain equivariant characteristic classes called Thom polynomials of multisingularities, established by Kazarian. Suitable substitutions into these polynomials completely characterize the hierarchy. As a consequence, the computation of degeneration posets becomes algorithmic in nature. In our second main result, we prove that our singularity-theoretic hierarchy extends the algebraic hierarchy obtained from deformation theory. While deformation theory requires the dimension (rank) to be fixed, our hierarchy generalizes this framework by comparing algebras of varying dimensions.
