{
  "id": "2111.06351",
  "version": "v1",
  "published": "2021-11-11T18:03:43.000Z",
  "updated": "2021-11-11T18:03:43.000Z",
  "title": "GIT stability of linear maps on projective space with marked points",
  "authors": [
    "Max Weinreich"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AG",
    "math.DS"
  ],
  "abstract": "We construct moduli spaces of linear self-maps of $N$-dimensional projective space with $n$ marked points, up to projective equivalence. That is, we let the special linear group act simultaneously by conjugation on projective linear maps and diagonally on $(\\mathbb{P}^N)^n$, and we take the geometric invariant theory (GIT) quotient. These moduli spaces arise in algebraic dynamics in two ways: first, as ambient varieties of degree 1 portrait spaces; second, as the domains of discrete integrable systems such as the pentagram map. Our main result is a dynamical characterization of the GIT semistable and stable loci in the space of linear maps with marked points. The proof is combinatorial: to describe the weight polytopes for this action, we compute the vertices and facets of certain convex polyhedra generated by roots of the $A_N$ lattice.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-11-11T18:03:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P45",
      "14L24",
      "52B05"
    ],
    "keywords": [
      "linear maps",
      "marked points",
      "projective space",
      "git stability",
      "special linear group act"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}