{
  "id": "2201.00033",
  "version": "v2",
  "published": "2021-12-31T19:18:17.000Z",
  "updated": "2022-11-09T19:34:14.000Z",
  "title": "Weierstrass semigroups from cyclic covers of hyperelliptic curves",
  "authors": [
    "Ethan Cotterill",
    "Nathan Pflueger",
    "Naizhen Zhang"
  ],
  "comment": "31 pages. Revised version now includes explicit effective calculations for a distinguished class of \"staircase\" semigroups; submitted",
  "categories": [
    "math.AG",
    "math.CO",
    "math.NT"
  ],
  "abstract": "The {\\it Weierstrass semigroup} of pole orders of meromorphic functions in a point $p$ of a smooth algebraic curve $C$ is a classical object of study; a celebrated problem of Hurwitz is to characterize which semigroups ${\\rm S} \\subset \\mathbb{N}$ with finite complement are {\\it realizable} as Weierstrass semigroups ${\\rm S}= {\\rm S}(C,p)$. In this note, we establish realizability results for cyclic covers $\\pi: (C,p) \\rightarrow (B,q)$ of hyperelliptic targets $B$ marked in hyperelliptic Weierstrass points; and we show that realizability is dictated by the behavior under $j$-fold multiplication of certain divisor classes in hyperelliptic Jacobians naturally associated to our cyclic covers, as $j$ ranges over all natural numbers.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-11-09T19:34:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H10",
      "14H30",
      "14H40",
      "14H45",
      "14H51",
      "14Gxx"
    ],
    "keywords": [
      "cyclic covers",
      "weierstrass semigroup",
      "hyperelliptic curves",
      "smooth algebraic curve",
      "hyperelliptic weierstrass points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}