{
  "id": "2408.14411",
  "version": "v1",
  "published": "2024-08-26T17:00:01.000Z",
  "updated": "2024-08-26T17:00:01.000Z",
  "title": "Positivity of the tangent bundle of rational surfaces with nef anticanonical divisor",
  "authors": [
    "Hosung Kim",
    "Jeong-Seop Kim",
    "Yongnam Lee"
  ],
  "comment": "20 pages, 14 figures",
  "categories": [
    "math.AG"
  ],
  "abstract": "In this paper, we study the property of bigness of the tangent bundle of a smooth projective rational surface with nef anticanonical divisor. We first show that the tangent bundle $T_S$ of $S$ is not big if $S$ is a rational elliptic surface. We then study the property of bigness of the tangent bundle $T_S$ of a weak del Pezzo surface $S$. When the degree of $S$ is $4$, we completely determine the bigness of the tangent bundle through the configuration of $(-2)$-curves. When the degree $d$ of $S$ is less than or equal to $3$, we get a partial answer. In particular, we show that $T_S$ is not big when the number of $(-2)$-curves is less than or equal to $7-d$, and $T_S$ is big when $d=3$ and $S$ has the maximum number of $(-2)$-curves. The main ingredient of the proof is to produce irreducible effective divisors on $\\mathbb{P}(T_S)$, using Serrano's work on the relative tangent bundle when $S$ has a fibration, or the total dual VMRT associated to a conic fibration on $S$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-26T17:00:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tangent bundle",
      "nef anticanonical divisor",
      "weak del pezzo surface",
      "positivity",
      "rational elliptic surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}