{
  "abstract": "The study of super Riemann surfaces and their moduli led Witten and Felder, Kazhdan and Polishchuk to ask what is the smallest $d$ such that the $d$-th power of the Schottky ideal is contained in the super Schottky ideal. Felder, Kazhdan and Polishchuk proved that $d=g$ in odd genus $g\\geq5$ and that $d\\in\\{g-1,g\\}$ in even genus $g\\geq4$; Y.~Shen subsequently proved that $d=g$ in even genus $g\\geq6$. Thus the only remaining open case was $g=4$. Here we close this gap by showing that in genus $4$ too, the smallest power of the Schottky ideal contained in the super Schottky ideal is $d=g=4$. The question is equivalent to one concerning the codifferential of the super period map. The quadratic odd contribution sends a conormal vector to a bivector, represented by a skew-symmetric $(2g-2)\\times(2g-2)$ matrix. The question is to find such a vector for which, at a generic point, this matrix has maximal rank $2g-2$, or $6$ in our case. Our computation uses the standard exact sequences on $C\\times C$ attached to the sheaves $\\OO(a,b,c)$, together with a degeneration to a vanishing theta-null and some facts about the Szegő kernel. At a curve with a vanishing theta-null, a regularized version of the codifferential can be described by an explicit multiplication map, allowing an easy computation of the rank. This rank turns out to be $4$. A first-order calculation on an explicit deformation of a cyclic trigonal vanishing theta-null differentiates the global Gaussian-map identity for the regularized conormal bivector. Including the variation of the Gaussian map gives a nonzero first normal symbol. The resulting first-order form is nonzero on the null space of the limiting skew-symmetric matrix, so the rank jumps to $6$ on nearby curves, showing that $d=4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16131",
  "author_total": 2,
  "authors": [
    {
      "name": "Ron Donagi"
    },
    {
      "name": "Simone Noja"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0ay3tqc9360bj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16131",
  "fulltext_url": null,
  "id": "0ay3tqc9360bj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16131",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213421990",
  "pdf_url": "https://arxiv.org/pdf/2609.16131",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0ay3tqc9360bj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:41Z",
      "native_id": "2609.16131",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16131"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Schottky versus super Schottky in genus 4",
  "title_text": "Schottky versus super Schottky in genus 4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/0ay3tqc9360bj",
  "venue": "arXiv",
  "versions": []
}