2-step ideals and commuting matrices

Source abstract

2-step ideals are ideals $I$ of the polynomial ring that satisfy $\mathfrak{m}^{k+2}\subsetneq I\subsetneq \mathfrak{m}^k$ where $\mathfrak{m}$ is the maximal ideal generated by variables. This class of ideals was recently introduced in [F. Giovenzana, L. Giovenzana, M. Graffeo, P. Lella: New components of Hilbert schemes of points and 2-step ideals, 2025, arxiv: 2507.02789] with the aim to obtain new irreducible components of $\mathrm{Hilb}^d(\mathbb{A}^n)$ and in particular to obtain new loci in $\mathrm{Hilb}^d(\mathbb{A}^3)$ of large dimension. In this paper we use the correspondence between Hilbert schemes and varieties of commuting matrices to define loci of commuting matrices that correspond to 2-step ideals. Then we estimate the dimensions of the obtained loci to get new proofs for the estimates of dimensions of loci of 2-step ideals in the case $n=3$. In this case we are also able to omit some assumptions in the dimension estimates in the above mentioned paper.