{
  "id": "2401.03821",
  "version": "v1",
  "published": "2024-01-08T11:19:10.000Z",
  "updated": "2024-01-08T11:19:10.000Z",
  "title": "On the degree of irrationality of low genus $K3$ surfaces",
  "authors": [
    "Federico Moretti",
    "Andrés Rojas"
  ],
  "comment": "30 pages, 1 figure. Comments welcome!",
  "categories": [
    "math.AG"
  ],
  "abstract": "Given a general polarized $K3$ surface $S\\subset \\mathbb P^g$ of genus $g\\le 14$, we study projections $S\\hookrightarrow \\mathbb P^g\\dashrightarrow \\mathbb P^2$ of minimal degree and their variational structure. In particular, we prove that the degree of irrationality of all such surfaces is at most $4$, and that for $g=7,8,9,11$ there are no rational maps $S\\dashrightarrow \\mathbb P^2$ of degree $3$ induced by the primitive linear system. Our methods combine vector bundle techniques \\`a la Lazarsfeld with derived category tools, and also make use of the rich theory of singular curves on $K3$ surfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-08T11:19:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E05",
      "14F08",
      "14J28"
    ],
    "keywords": [
      "low genus",
      "irrationality",
      "vector bundle techniques",
      "rich theory",
      "study projections"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}