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2-step ideals and commuting matrices

  • Klemen Šivic
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Abstract

2-step ideals are ideals I of the polynomial ring that satisfy mk+2subsetneq Isubsetneq mk where 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 Hilbd(𝔸n) and in particular to obtain new loci in Hilbd(𝔸3) 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.

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Cite

BibTeXRIS
BibTeX
@misc{sivic20262step,
  title = {{2-step ideals and commuting matrices}},
  author = {Klemen Šivic},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24491},
  eprint = {2609.24491},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24491},
}
RIS
TY  - UNPB
TI  - 2-step ideals and commuting matrices
AU  - Klemen Šivic
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24491
UR  - https://arxiv.org/abs/2609.24491
AB  - 2-step ideals are ideals I of the polynomial ring that satisfy m^k+2subsetneq Isubsetneq m^k where 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 Hilb^d(𝔸^n) and in particular to obtain new loci in Hilb^d(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.
ER  -