{
  "abstract": "Let $π: X\\to \\mathbb P^1$ be a semistable Jacobian elliptic surface over $\\mathbb C$, and set $χ=χ(\\mathcal O_X)\\ge3$, so that $κ(X)=1$. Assume that the Mordell-Weil group of $π$ is finite and that $π$ has at least one reducible fiber, the reducible fibers being of types $I_{n_1},\\cdots, I_{n_s}$. Recently, Laface et al. proved that the zero section and the components of the reducible fibers generate $\\overline{\\mathrm NE}(X)$ if and only if $$δ(π):=\\sum_{i=1}^s\\frac{\\lfloor n_i^2/4\\rfloor}{n_i}\\leχ.$$ In particular, the Mori cone is rational polyhedral in this range. They also proved that $N(π)=\\sum_{i=1}^s n_i\\le 2χ+3$ implies that $X$ is a Mori dream surface. In this paper, we study the existence problem of Mori dream surfaces provided that $N(π)\\ge 2χ+4$. Suppose $δ(π)\\le χ$. We first show that every nef isotropic divisor on $X$ is semiample, and whenever each $n_i$ is even. Furthermore, when every $n_i$ is even, we obtain criteria for $X$ to be a Mori dream surface: $X$ is a Mori dream surface provided that either (i) all $n_i=2$, or (ii) $N(π)\\le 2χ+4$; if some $n_j=4$, then $X$ is a Mori dream surface if and only if $N(π)\\le 2χ+4$. However, once some $n_i$ is odd, we construct a Jacobian elliptic surface $π: Y\\to\\mathbb P^1$ with $δ(π)=χ=3$, trivial Mordell-Weil group, and singular-fiber configuration $I_4+3I_3+23I_1$ for which none of the eight non-vertical isotropic extremal rays is semiample. In particular, $Y$ is not a Mori dream surface.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23306",
  "author_total": 2,
  "authors": [
    {
      "name": "Antonio Laface"
    },
    {
      "name": "Sichen Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5ge3k0qwv9x78.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23306",
  "fulltext_url": null,
  "id": "5ge3k0qwv9x78",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23306",
  "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": "W7214041019",
  "pdf_url": "https://arxiv.org/pdf/2609.23306",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5ge3k0qwv9x78.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23306",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23306"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Semiampleness on Jacobian elliptic surfaces of Kodaira dimension one",
  "title_text": "Semiampleness on Jacobian elliptic surfaces of Kodaira dimension one",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/5ge3k0qwv9x78",
  "venue": "arXiv",
  "versions": []
}