{
  "id": "2406.12166",
  "version": "v1",
  "published": "2024-06-18T00:36:12.000Z",
  "updated": "2024-06-18T00:36:12.000Z",
  "title": "Universal polynomials for multi-singularity loci of maps",
  "authors": [
    "Toru Ohmoto"
  ],
  "comment": "79 pp",
  "categories": [
    "math.AG"
  ],
  "abstract": "In the present paper, we prove the existence of universal polynomials which express multi-singularity loci classes of prescribed types for proper morphisms between smooth schemes over an algebraically closed field of characteristic zero -- we call them Thom polynomials for multi-singularity types of maps. It has been referred to as the Thom-Kazarian principle and unsolved for a long time. This result solidifies the foundation for a general enumerative theory of singularities of maps which is applicable to a broad range of problems in classical and modern algebraic geometry. In particular, it would contribute to a satisfactory answer to the rest of (an advanced form of) Hilbert's 15th problem and connect such classics to recent new interests in enumerations inspired by mathematical physics and other fields. A main feature of our proof is a striking use of algebro-geometric cohomology operations. Somewhat surprisingly, when trying to grasp a full perspective of classical enumerative geometry, we will inevitably encounter algebraic cobordism and motivic cohomology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-18T00:36:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "universal polynomials",
      "express multi-singularity loci classes",
      "inevitably encounter algebraic cobordism",
      "algebro-geometric cohomology operations",
      "hilberts 15th problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 79,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}