{
  "id": "intersection-numbers-and-positivity",
  "title": "Intersection numbers and positivity",
  "description": "Intersection numbers of line bundles on proper schemes, defined through Snapper's polynomial of Euler characteristics, with multilinearity, reduction to components and divisors, degrees on curves, pullback and positivity for ample bundles; the Nakai-Moishezon criterion for ampleness; nef line bundles and Kleiman's theorem that they have nonnegative intersection numbers, so that nef plus ample is ample; and Bézout's inequality for the isolated zeros of n polynomials in n variables, with a weighted version.",
  "units": [
    {
      "id": "intersection-numbers-of-line-bundles",
      "title": "Intersection numbers of line bundles",
      "reader": "intersection-numbers-of-line-bundles.html",
      "source": "src/intersection-numbers-of-line-bundles.md",
      "source_sha256": "882a0db424a565db7d1ae7266648d30884d02ab1329be4b72c5142caf5c1d996",
      "status": "author_self_checked"
    },
    {
      "id": "the-nakai-moishezon-criterion",
      "title": "The Nakai–Moishezon criterion",
      "reader": "the-nakai-moishezon-criterion.html",
      "source": "src/the-nakai-moishezon-criterion.md",
      "source_sha256": "16563c85c6ab4263cb82c253e85c97f8265fb9b71db779176716dfcbff2be2ec",
      "status": "author_self_checked"
    },
    {
      "id": "nef-line-bundles-and-kleimans-theorem",
      "title": "Nef line bundles and Kleiman's theorem",
      "reader": "nef-line-bundles-and-kleimans-theorem.html",
      "source": "src/nef-line-bundles-and-kleimans-theorem.md",
      "source_sha256": "624ac424b4c845d9baf688e4de000ca88b819550d7cc4a961148fc663dc8d959",
      "status": "author_self_checked"
    },
    {
      "id": "bezouts-inequality-for-isolated-zeros",
      "title": "Bézout's inequality for isolated zeros",
      "reader": "bezouts-inequality-for-isolated-zeros.html",
      "source": "src/bezouts-inequality-for-isolated-zeros.md",
      "source_sha256": "fedaa23f8ebe9b096889aa6c0f724ccf725da31be2770bf62a44da60b0f47eca",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "AG-QC lessons on Euler characteristics and Hilbert polynomials, coherent direct images, and Serre's theorems",
    "AG-MO lessons on ample and very ample sheaves, dimension of fibres, and Zariski's Main Theorem",
    "AG-CA lessons on completion, dimension theory, and Cohen-Macaulay rings"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}