{
  "id": "2108.01506",
  "version": "v1",
  "published": "2021-08-03T13:47:54.000Z",
  "updated": "2021-08-03T13:47:54.000Z",
  "title": "Moduli spaces of quasitrivial sheaves on the three dimensional projective space",
  "authors": [
    "Douglas Guimarães",
    "Marcos Jardim"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "A torsion free sheaf $E$ on $\\mathbb{P}^d$ is called \\emph{quasitrivial} if $E^{\\vee\\vee}=\\mathcal{O}_{\\mathbb{P}^3}^{\\oplus r}$ and $\\dim(E^{\\vee\\vee}/E)=0$. While such sheaves are always $\\mu$-semistable, they may not be semistable. We study the Gieseker--Maruyama moduli space $\\mathcal{N}(r,n)$ of rank $r$ semistable quasitrivial sheaves on $\\mathbb{P}^3$ with $h^0(E^{\\vee\\vee}/E)=n$ via the Quot scheme of points $Quot(\\mathcal{O}_{\\mathbb{P}^3}^{\\oplus r},n)$. We show that $\\mathcal{N}(r,n)$ is empty when $r>n$, while $\\mathcal{N}(n,n)$ has no stable points and is isomorphic to the symmetric product $Sym^n(\\mathbb{P}^3)$. Our main result is the construction of an irreducible component of $\\mathcal{N}(r,n)$ of dimension $2n+rn-r^2+1$ when $r<n$. Furthermore, this is the only irreducible component when $n\\le10$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-03T13:47:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dimensional projective space",
      "irreducible component",
      "gieseker-maruyama moduli space",
      "torsion free sheaf",
      "quot scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}