{
  "abstract": "For a point $P$ on a smooth projective curve $C$ of genus $g$, the section ring $R_d = R(C,\\mathcal{O}_C(dP))$ can be minimally presented as a quotient $S_d/I_d$ where $S_d$ is a $\\mathbb{Z}$-graded polynomial ring. Motivated by Green's $N_p$ properties for projective embeddings, we investigate the syzygies of $R_d$ over $S_d$ in low degrees $d$ when $R_d$ is not generated in degree 1. We bound the degrees of the generators of $R_d$ and prove uniform column-by-column bounds on the support of the Betti table of $R_d$ over $S_d$. We compute the weighted regularity of $R_d$ and show that if $d$ is larger than the Frobenius number of $P$ then $R_d$ satisfies the weighted $N_p$ condition, where $p=g-1-\\binom{d-g}{2}$. Finally, we give sufficient criteria for the Betti numbers to be determined explicitly and show that for ordinary points, the resolution of $R_{g+1}$ is pure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00312",
  "author_total": 3,
  "authors": [
    {
      "name": "Maya Banks"
    },
    {
      "name": "John Cobb"
    },
    {
      "name": "Mahrud Sayrafi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ef8w1q8cc9d8k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00312",
  "fulltext_url": null,
  "id": "ef8w1q8cc9d8k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00312",
  "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"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7206168854",
  "pdf_url": "https://arxiv.org/pdf/2609.00312",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ef8w1q8cc9d8k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00312",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00312"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Weighted Syzygies of Pointed Curves",
  "title_text": "Weighted Syzygies of Pointed Curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/ef8w1q8cc9d8k",
  "venue": "arXiv",
  "versions": []
}