{
  "id": "2307.16272",
  "version": "v1",
  "published": "2023-07-30T16:39:00.000Z",
  "updated": "2023-07-30T16:39:00.000Z",
  "title": "Murphy's law on a fixed locus of the Quot scheme",
  "authors": [
    "Reinier F. Schmiermann"
  ],
  "comment": "24 pages, comments are welcome",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $T := \\mathbb{G}_m^d$ be the torus acting on the Quot scheme of points $\\coprod_n \\mathrm{Quot}_{\\mathcal{O}^r/\\mathbb{A}^d/\\mathbb{Z}}^n$ via the standard action on $\\mathbb{A}^d$. We analyze the fixed locus of the Quot scheme under this action. In particular we show that for $d \\leq 2$ or $r \\leq 2$, this locus is smooth, and that for $d \\geq 4$ and $r \\geq 3$ it satisfies Murphy's law as introduced by Vakil, meaning that it has arbitrarily bad singularities. These results are obtained by giving a decomposition of the fixed locus into connected components, and identifying the components with incidence schemes of subspaces of $\\mathbb{P}^{r-1}$. We then obtain a characterization of the incidence schemes which occur, in terms of their graphs of incidence relations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-30T16:39:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D20",
      "14B05",
      "14M25"
    ],
    "keywords": [
      "quot scheme",
      "fixed locus",
      "incidence schemes",
      "satisfies murphys law",
      "arbitrarily bad singularities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}