{
  "id": "2110.03917",
  "version": "v3",
  "published": "2021-10-08T06:22:32.000Z",
  "updated": "2022-03-03T05:54:46.000Z",
  "title": "On behavior of conductors, Picard schemes, and Jacobian numbers of varieties over imperfect fields",
  "authors": [
    "Ippei Nagamachi",
    "Teppei Takamatsu"
  ],
  "comment": "35 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a regular geometrically integral variety over an imperfect field $K$. Unlike the case of characteristic $0$, $X':=X\\times_{\\mathrm{Spec}\\,K}\\mathrm{Spec}\\,K'$ may have singular points for a (necessarily inseparable) field extension $K'/K$. In this paper, we define new invariants of the local rings of codimension $1$ points of $X'$, and use these invariants for the calculation of $\\delta$-invariants (, which relate to genus changes,) and conductors of such points. As a corollary, we give refinements of Tate's genus change theorem and the Patakfalvi-Waldron Theorem. Moreover, when $X$ is a curve, we show that the Jacobian number of $X$ is $2p/(p-1)$ times of the genus change by using the above calculation. In this case, we also relate the structure of the Picard scheme of $X$ with invariants of singular points of $X$. To prove such a relation, we give a characterization of the geometrical normality of algebras over fields of positive characteristic.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2022-03-03T05:54:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H20",
      "14G17",
      "14K30"
    ],
    "keywords": [
      "picard scheme",
      "imperfect field",
      "jacobian number",
      "conductors",
      "singular points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}