{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24491",
  "author_total": 1,
  "authors": [
    {
      "name": "Klemen Šivic"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6m9nr67wy41bm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24491",
  "fulltext_url": null,
  "id": "6m9nr67wy41bm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24491",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7214048015",
  "pdf_url": "https://arxiv.org/pdf/2609.24491",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6m9nr67wy41bm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24491",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24491"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "2-step ideals and commuting matrices",
  "title_text": "2-step ideals and commuting matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/6m9nr67wy41bm",
  "venue": "arXiv",
  "versions": []
}