{
  "id": "1510.04604",
  "version": "v1",
  "published": "2015-10-15T16:10:47.000Z",
  "updated": "2015-10-15T16:10:47.000Z",
  "title": "Intersection Theory on Tropicalizations of Toroidal Embeddings",
  "authors": [
    "Andreas Gross"
  ],
  "comment": "Comments are welcome!",
  "categories": [
    "math.AG"
  ],
  "abstract": "We show how to equip the cone complexes of toroidal embeddings with additional structure that allows to define a balancing condition for weighted subcomplexes. We then proceed to develop the foundations of an intersection theory on cone complexes including push-forwards, intersections with tropical divisors, and rational equivalence. These constructions are shown to have an algebraic interpretation: Ulirsch's tropicalizations of subvarieties of toroidal embeddings carry natural multiplicities making them tropical cycles, and the induced tropicalization map for cycles respects push-forwards, intersections with boundary divisors, and rational equivalence. As an application we prove a correspondence between the genus 0 tropical descendant Gromov-Witten invariants introduced by Markwig and Rau and the genus 0 logarithmic descendant Gromov-Witten invariants of toric varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-15T16:10:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14T05"
    ],
    "keywords": [
      "intersection theory",
      "tropicalization",
      "toroidal embeddings carry natural multiplicities",
      "logarithmic descendant gromov-witten invariants",
      "rational equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151004604G"
    }
  }
}