Deformations of local Artin rings via Hilbert-Burch matrices

TL;DR

This paper explores the structure of local Artin rings through Hilbert-Burch matrices, providing explicit parametrizations of Gr"obner cells and analyzing deformations that preserve the Hilbert function.

Contribution

It introduces a method to compute Betti strata and deformations of ideals in local rings using Hilbert-Burch matrices, advancing the understanding of local Hilbert schemes.

Findings

Explicit parametrization of local Gr"obner cells

Computation of Betti strata in local Artin rings

Examples of deformations preserving the Hilbert function

Abstract

In the local setting, Gr\"obner cells are affine spaces that parametrize ideals in $\mathbf{k}[\![x,y]\!]$ that share the same leading term ideal with respect to a local term ordering. In particular, all ideals in a cell have the same Hilbert function, so they provide a cellular decomposition of the punctual Hilbert scheme compatible with its Hilbert function stratification. We exploit the parametrization given in \cite{HW21} via Hilbert-Burch matrices to compute the Betti strata, with hands-on examples of deformations that preserve the Hilbert function, and revisit some classical results along the way. Moreover, we move towards an explicit parametrization of all local Gr\"obner cells.