{
  "abstract": "Let $S=S_{2,\\ldots,2}\\subset\\mathbb P^{n+2}$ be a smooth complete intersection of $n$ quadrics containing a line $E$, where $n\\geq 3$, and set $D=(n-1)\\mathbf h+E$, where $\\mathbf h=[\\mathcal O_S(1)]$. We study the deformations of smooth curves $C\\in |mD|$ and their obstructions. The linear system $|D|$ defines a birational morphism $f:S\\to\\overline S$ that contracts $E$ to a point and realizes an unprojection in the sense of Papadakis--Reid. When $n=3$, the surface $S=S_{2,2,2}\\subset\\mathbb P^5$ is a $K3$ surface and $E$ is a $(-2)$-curve. We prove that the complementary open surface $S\\setminus E$ admits first-order infinitesimal deformations in $\\mathbb P^5\\setminus E$ that are obstructed at the second order. Using these obstructions, we show that, for every sufficiently large $m$ with $\\operatorname{char} k\\nmid m$, there exists a generically non-reduced irreducible component $W_m$ of the Hilbert scheme $\\operatorname{Hilb}^{sc}\\mathbb P^5$ of smooth connected curves in $\\mathbb P^5$. Its general member $C$ lies on a complete intersection $S_{2,2,2}$ of three quadrics, satisfies $C\\sim mD$, and has $h^0(N_{C/\\mathbb P^5})-\\dim_{[C]}\\operatorname{Hilb}^{sc}\\mathbb P^5=3$. In contrast, when $n>3$, we prove that, for every sufficiently large $m$ with $\\operatorname{char} k\\nmid m$, the corresponding family of curves in $|mD|$ gives rise to a generically smooth irreducible component of $\\operatorname{Hilb}^{sc}\\mathbb P^{n+2}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30962",
  "author_total": 1,
  "authors": [
    {
      "name": "Hirokazu Nasu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5hnrmg6vxfp1h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30962",
  "fulltext_url": null,
  "id": "5hnrmg6vxfp1h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30962",
  "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": "W7223374338",
  "pdf_url": "https://arxiv.org/pdf/2610.30962",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5hnrmg6vxfp1h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30962",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30962"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Obstructions to deforming space curves lying on a complete intersection of quadrics",
  "title_text": "Obstructions to deforming space curves lying on a complete intersection of quadrics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/5hnrmg6vxfp1h",
  "venue": "arXiv",
  "versions": []
}