{
  "abstract": "Recently, Huang and Satriano determined the sharp dimension threshold for Batyrev's non-negativity conjecture on stringy Hodge numbers: the conjecture holds in dimensions at most four and fails in every dimension at least five. We show that their counterexamples fit into a single broad three-parameter family $Y_{m,d}:=V(\\sum_{i=0}^m s_i g_i(x)+h(x))\\subset\\mathbb{P}^{m+d+1}$, where each $g_i$ is homogeneous of degree $d-1$ and $h$ is homogeneous of degree $d$, whose singular locus is a linear space $Λ\\cong\\mathbb{P}^m$. Blowing up $Λ$ gives a log resolution with one smooth exceptional divisor of discrepancy one. We compute the Hodge-Deligne polynomials of both the resolution and the exceptional divisor from their projective-space fibrations over $\\mathbb{P}^d$, derive a master formula for $E_{\\mathrm{st}}(Y_{m,d}\\times(\\mathbb{P}^1)^n)$, and prove the structure formula $h_{\\mathrm{st}}^{p,q}(Y_{m,d}\\times(\\mathbb{P}^1)^n)=A_{p,q}+B_{p,q}-C_{p,q}$, where the three terms are non-negative. Thus the only possible source of negativity is $-C_{p,q}$, which is governed by the primitive middle cohomology of $Z_{m,d}=V(g_0,\\ldots,g_m)\\subset\\mathbb{P}^d$. Off the diagonal, every negative entry equals $-C_{p,q}$, a binomially weighted sum of shifted primitive Hodge numbers of $Z_{m,d}$. On the diagonal, the behaviour is governed by the parity of $m+d$. We also obtain a complete classification for $0\\le m\\le d-1$, $d\\ge3$, and $n\\ge1$. The only non-counterexamples are $(m,d)=(0,3)$ for every $n\\ge1$, and $(m,d)=(2,3)$ for $n\\ge4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06829",
  "author_total": 1,
  "authors": [
    {
      "name": "Dimitrios I. Dais"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33e4bxw088rdw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06829",
  "fulltext_url": null,
  "id": "33e4bxw088rdw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06829",
  "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": "W7212068406",
  "pdf_url": "https://arxiv.org/pdf/2609.06829",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33e4bxw088rdw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.06829",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06829"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A unified family of counterexamples to Batyrev's non-negativity conjecture on stringy Hodge numbers",
  "title_text": "A unified family of counterexamples to Batyrev's non-negativity conjecture on stringy Hodge numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/33e4bxw088rdw",
  "venue": "arXiv",
  "versions": []
}