{
  "abstract": "Given a point $[(\\widetilde{C},C)]$ of the moduli space ${\\mathcal R}_g$, where $\\widetilde{C}$ is an étale double cover of a smooth curve $C$ of genus $g \\,\\geq\\, 7$, we construct a twisted meromorphic bidifferential $\\widehatη_P$ on $C\\times C$ with a pole on the diagonal. It is shown that the second fundamental form $ρ_P$ of the Prym map from ${\\mathcal R}_g$ to ${\\mathcal A}_{g-1}$ at $[(\\widetilde{C},C)]$ is given by this $\\widehatη_P$. This allows us to compare $ρ_P$ with the second fundamental forms of the two Torelli maps, to ${\\mathcal A}_{2g-1}$ and ${\\mathcal A}_{g}$, at the points $[\\widetilde{C}]$ and $[C]$ respectively. It is shown that $\\widehatη_P$ induces a projective structure on $C$ that depends on the double cover $\\widetilde{C}\\, \\longrightarrow\\, C$; these projective structures together give a ${\\mathcal C}^{\\infty}$ section $β_P$ of the bundle of projective structures on the moduli space ${\\mathcal R}_g$. Finally, it is proved that the $(0,\\,1)$-component of the differential of the section $β_P$ is the pullback of the Kähler form of the Siegel metric on ${\\mathcal A}_{g-1}$ via the Prym map. This is a key ingredient in the proof that this section $β_P$ is not the pullback of a section of projective structures on ${\\mathcal M}_{g}$ and, in addition, that it differs from the known projective structures on ${\\mathcal R}_g$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28156",
  "author_total": 4,
  "authors": [
    {
      "name": "Indranil Biswas"
    },
    {
      "name": "Elisabetta Colombo"
    },
    {
      "name": "Paola Frediani"
    },
    {
      "name": "Alessandro Ghigi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ee0mf5gpberv8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28156",
  "fulltext_url": null,
  "id": "ee0mf5gpberv8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28156",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223151525",
  "pdf_url": "https://arxiv.org/pdf/2610.28156",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ee0mf5gpberv8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28156",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28156"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Twisted bidifferential, second fundamental form of Prym map and projective structures",
  "title_text": "A Twisted bidifferential, second fundamental form of Prym map and projective structures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/ee0mf5gpberv8",
  "venue": "arXiv",
  "versions": []
}