{
  "abstract": "Let $F$ be a reduced plane projective curve defined over an algebraically closed field $k$. The Bourbaki degree of $F$, denoted by $\\mathrm{Bour}(F)$, measures its freeness and was introduced in \\cite{JNS2024}. It is defined as the degree of $R/I_ε$, where $R = k[x,y,z]$ and $I_ε\\subseteq R$ is the Bourbaki ideal associated with a minimal generator $ε$ of the module of first syzygies of the Jacobian ideal $J_F$. In this article, we investigate the local Bourbaki degree and apply it to refine the well-known upper bound for the global Bourbaki degree. We then study curves with prescribed Bourbaki degree through the minimal graded free resolution of $R/J_F$, describing the possible resolution forms and expressing $\\mathrm{Bour}(F)$ in terms of the corresponding graded shifts. Finally, we study the realization problem in fixed degree. We prove that for $°F \\geq 5$, the du Plessis-Wall bounds impose no numerical obstruction to the realization of the Bourbaki degree. Moreover, we show that every possible Bourbaki degree is realized by reduced plane curves of degrees $5$ and $6$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.06956",
  "author_total": 3,
  "authors": [
    {
      "name": "Roberto Alvarenga"
    },
    {
      "name": "Murillo Lozano"
    },
    {
      "name": "Parham Salehyan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8v7g5hcnzpk5z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.06956",
  "fulltext_url": null,
  "id": "8v7g5hcnzpk5z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.06956",
  "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.AG"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7160869638",
  "pdf_url": "https://arxiv.org/pdf/2605.06956",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8v7g5hcnzpk5z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2605.06956",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.06956"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Bourbaki Degree of Plane Projective Curves",
  "title_text": "On the Bourbaki Degree of Plane Projective Curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/8v7g5hcnzpk5z",
  "venue": "arXiv",
  "versions": []
}