{
  "id": "2007.12855",
  "version": "v1",
  "published": "2020-07-25T04:39:31.000Z",
  "updated": "2020-07-25T04:39:31.000Z",
  "title": "Bounding cohomology on a smooth projective surface with Picard number 2",
  "authors": [
    "Sichen Li"
  ],
  "comment": "6 pages, a simplified version of arxiv:1805.10741. Comments welcome!",
  "categories": [
    "math.AG"
  ],
  "abstract": "The following conjecture arose out of discussions between B. Harbourne, J. Ro\\'e, C. Cilberto and R. Miranda: for a smooth projective surface $X$ there exists a positive constant $c_X$ such that $h^1(\\mathcal O_X(C))\\le c_X h^0(\\mathcal O_X(C))$ for every prime divisor $C$ on $X$. When the Picard number $\\rho(X)=2$, we prove that if either the Kodaira dimension $\\kappa(X)=1$ and $X$ has a negative curve or $X$ has two negative curves, then this conjecture holds for $X$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-25T04:39:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C20"
    ],
    "keywords": [
      "smooth projective surface",
      "picard number",
      "bounding cohomology",
      "negative curve",
      "conjecture arose"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}