{
  "id": "2002.11010",
  "version": "v1",
  "published": "2020-02-25T16:35:22.000Z",
  "updated": "2020-02-25T16:35:22.000Z",
  "title": "Bigness of the tangent bundle of del Pezzo surfaces and $D$-simplicity",
  "authors": [
    "Devlin Mallory"
  ],
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "We consider the question of simplicity of a ring $R$ under the action of its ring of differential operators $D_R$. We give examples to show that even when $R$ is Gorenstein and has rational singularities $R$ need not be a simple $D_R$-module; for example, this is the case when $R$ is the homogeneous coordinate ring of a smooth cubic surface. Our examples are homogeneous coordinate rings of smooth Fano varieties, and our proof proceeds by showing that the tangent bundle of such a variety need not be big. We also give a partial converse showing that when $R$ is the homogeneous coordinate ring of a smooth projective variety $X$, embedded by some multiple of its canonical divisor, then simplicity of $R$ as a $D_R$-module implies that $X$ is Fano and thus $R$ has rational singularities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-25T16:35:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13N10",
      "14B05",
      "14J60"
    ],
    "keywords": [
      "del pezzo surfaces",
      "tangent bundle",
      "homogeneous coordinate ring",
      "simplicity",
      "rational singularities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}