{
  "abstract": "Let $f:S\\to C$ be a relatively minimal semistable fibration of genus $g\\ge2$ over an algebraically closed field of characteristic $p>0$ with smooth and geometrically connected generic fiber. Let $Σ\\subset C$ denote the set of points over which the fibers are singular, with $s=|Σ|$. If the total surface $S$ is Hodge Witt and $s\\ge2$, we prove the classical Arakelov inequality $$ \\text{deg}\\, f_*ω_{S/C}\\le \\frac g2 \\text{deg}\\, Ω_C^1(\\logΣ), $$ where $ω_{S/C}$ denotes the relative dualizing sheaf. For $C=\\mathbb P^1$, we obtain the stronger estimate $$ \\text{deg}\\, f_*ω_{S/\\mathbb P^1}\\le \\frac g2(s-2)-\\frac12 b_1(S), $$ where $b_1(S)$ is the first Betti number of $S$. We also construct a genus-$2$ semistable fibration over $\\mathbb P^1$ with exactly $4$ singular fibers in characteristic $5$. Its total surface is Hodge Witt, and $$ \\text{deg}\\, f_*ω_{S/\\mathbb P^1}=2=\\frac g2(s-2). $$ Thus Nguyen's lower bound $s\\ge4$ and the Hodge--Witt Arakelov inequality are both sharp. As applications, we derive a canonical-class inequality and a Szpiro-type inequality with coefficients linear in $g$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02760",
  "author_total": 2,
  "authors": [
    {
      "name": "Hao Max Sun"
    },
    {
      "name": "Wan-Yuan Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a5mkwqct1b13q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02760",
  "fulltext_url": null,
  "id": "a5mkwqct1b13q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02760",
  "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": "W7220452910",
  "pdf_url": "https://arxiv.org/pdf/2610.02760",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a5mkwqct1b13q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02760",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02760"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Arakelov inequalities for fibered surfaces in positive characteristic",
  "title_text": "Arakelov inequalities for fibered surfaces in positive characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/a5mkwqct1b13q",
  "venue": "arXiv",
  "versions": []
}