{
  "id": "2206.04952",
  "version": "v1",
  "published": "2022-06-10T09:01:46.000Z",
  "updated": "2022-06-10T09:01:46.000Z",
  "title": "Classification and geometric properties of surfaces with property ${\\bf N}_{3,3}$",
  "authors": [
    "Hoang Le Truong"
  ],
  "comment": "23 pages. Comments welcome!",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a closed subscheme of codimension $e$ in a projective space. One says that $X$ satisfies property ${\\bf N}_{d,p}$, if the $i$-th syzygies of the homogeneous coordinate ring are generated by elements of degree $<d+i$ for $0\\le i\\le p$. The geometric and algebraic properties of smooth projective varieties satisfying property ${\\bf N}_{2,e}$ are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: projective surfaces in $\\Bbb P^5$ satisfying property ${\\bf N}_{3,3}$. In particular, we give a classification of such varieties using adjunction mappings and we also provide illuminating examples of our results via calculations done with Macaulay 2. As corollaries, we study the CI-biliaison equivalence class of smooth projective surfaces of degree $10$ satisfying property ${\\bf N}_{3,3}$ on a cubic fourfold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-10T09:01:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "geometric properties",
      "projective surfaces",
      "ci-biliaison equivalence class",
      "smooth projective varieties satisfying property",
      "th syzygies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}