Degenerations of multisingularities and Artin algebras

Abstract

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.