{
  "id": "1907.03330",
  "version": "v1",
  "published": "2019-07-07T18:32:12.000Z",
  "updated": "2019-07-07T18:32:12.000Z",
  "title": "Counting rational curves on K3 surfaces with finite group actions",
  "authors": [
    "Sailun Zhan"
  ],
  "comment": "36 pages. Comments are very welcome",
  "categories": [
    "math.AG"
  ],
  "abstract": "G\\\"ottsche gave a formula for the dimension of the cohomology of Hilbert schemes of points on a smooth projective surface $S$. When $S$ admits an action by a finite group $G$, we describe the action of $G$ on the cohomology spaces. In the case that $S$ is a K3 surface, each element of $G$ gives a trace on $\\sum_{n=0}\\sum_{i=0}(-1)^{i}H^{i}(S^{[n]},\\mathbb{C})q^{n}$. When $G$ acts faithfully and symplectically on $S$, the resulting generating function is of the form $q/f(q)$, where $f(q)$ is a cusp form. We relate the cohomology of Hilbert schemes of points to the cohomology of the compactified Jacobian of the tautological family of curves over an integral linear system on a K3 surface as $G$-representations. We give a sufficient condition for a $G$-orbit of curves with nodal singularities not to contribute to the representation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-07T18:32:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "k3 surface",
      "finite group actions",
      "counting rational curves",
      "cohomology",
      "hilbert schemes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}