{
  "id": "1904.02047",
  "version": "v1",
  "published": "2019-04-03T15:09:31.000Z",
  "updated": "2019-04-03T15:09:31.000Z",
  "title": "Sets of points which project to complete intersections",
  "authors": [
    "Luca Chiantini",
    "Juan Migliore"
  ],
  "comment": "The authors of the appendix are A. Bernardi, L. Chiantini, G. Denham, G. Favacchio, B. Harbourne, J. Migliore, T. Szemberg and J. Szpond. 23 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "The motivating problem addressed by this paper is to describe those non-degenerate sets of points $Z$ in $\\mathbb P^3$ whose general projection to a general plane is a complete intersection of curves in that plane. One large class of such $Z$ is what we call $(m,n)$-grids. We relate this problem to the {\\em unexpected cone property} ${\\mathcal C}(d)$, a special case of the unexpected hypersurfaces which have been the focus of much recent research. After an analysis of ${\\mathcal C}(d)$ for small $d$, we show that a non-degenerate set of $9$ points has a general projection that is the complete intersection of two cubics if and only if the points form a $(3,3)$-grid. However, in an appendix we describe a set of $24$ points that are not a grid but nevertheless have the projection property. These points arise from the $F_4$ root system. Furthermore, from this example we find subsets of $20$, $16$ and $12$ points with the same feature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-03T15:09:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M10",
      "14N20",
      "14N05",
      "14M07"
    ],
    "keywords": [
      "complete intersection",
      "general projection",
      "non-degenerate set",
      "large class",
      "general plane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}