{
  "id": "1606.03290",
  "version": "v1",
  "published": "2016-06-10T12:26:10.000Z",
  "updated": "2016-06-10T12:26:10.000Z",
  "title": "Linear systems on irregular varieties",
  "authors": [
    "Miguel Ángel Barja",
    "Rita Pardini",
    "Lidia Stoppino"
  ],
  "comment": "50 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a smooth complex projective variety of dimension $n$, let $a\\colon X\\rightarrow A$ be a map to an abelian variety such that $\\dim a(X)=\\dim X$ and the induced map $\\mathrm{Pic}^0(A)\\to \\mathrm{Pic}^0(X)$ is injective, and let $L\\in \\mathrm{Pic}(X)$; denote by $X^{(d)}\\to X$ the connected \\'etale cover induced by the $d$-th multiplication map of $A$ and by $L^{(d)}$ the pull-back of $L$ to $X^{(d)}$. We study the linear systems $|L^{(d)}\\otimes \\alpha|$ where $\\alpha$ is the pull back of an element of $\\mathrm{Pic}^0(A)$. When these systems are non empty for $\\alpha$ general, we prove that there exists a map $\\varphi\\colon X\\to Z$ such that $a$ factorizes through $\\varphi$ and induces by base change, for $d>>0$ and $\\alpha$ general, the map $\\varphi^{(d)}$ given by the linear systems $|L^{(d)}\\otimes \\alpha|$. Let $h^0_a(L)$ be the continuous rank of $L$, i.e., the generic value of $h^0(L\\otimes\\alpha)$ for $\\alpha\\in a^*\\mathrm{Pic}^0(A)$. We prove that the function $x\\mapsto h^0_a(L+xM)$, where $M$ is the pull back of a fixed very ample $H\\in \\mathrm{Pic}(A)$, extends to a convex differentiable function $\\phi(x)$ on $\\mathbb R$. We investigate thoroughly the regularity of $\\phi$. From these results we also deduce new differentiability properties of the volume function in our setting. Thanks to these results we prove various Clifford-Severi type inequalities of the form $\\mathrm{vol}(L)\\ge C(n) h^0_a(L)$, extending and refining previous results of the first named author. We also prove Castelnuovo type inequalities, of the form $h^0_a(kL)\\ge C(k,n)h^0_a(L)$, where $C(k,n)=\\mathcal O(k^n)$. Finally we characterize the triples $(X,a,L)$ in the limit cases of the Clifford-Severi inequalities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-10T12:26:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C20",
      "14J29",
      "14J30",
      "14J35",
      "14J40"
    ],
    "keywords": [
      "linear systems",
      "irregular varieties",
      "smooth complex projective variety",
      "castelnuovo type inequalities",
      "clifford-severi type inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}