{
  "annotation_id": "26931bd1-fa98-466e-85a6-81f3b667274d",
  "date_created": "2026-02-17T05:53:04.823000Z",
  "date_modified": "2026-02-17T05:53:04.823000Z",
  "file_hash": "4035a378dd04587a85246a203a4d2cbcf30714c324e7537dd69f4f68a8374819",
  "private": false,
  "record": {
    "abstract": "For a smooth, projective scheme $X$ over a field $k$ or any variety $X$ if $k$ has characteristic zero, we compute the compactly supported $\\mathbb{A}^1$-Euler characteristic of $\\operatorname{Sym}^2(X)$ if $\\operatorname{char}(k) \\ne 2$ and of $\\operatorname{Sym}^3(X)$ if $\\operatorname{char}(k) \\ne 2,3$. We do so by extending the definition of a $G$-equivariant quadratic Euler characteristic first studied by Pajwani-P\\\u0027al to arbitrary characteristic and by studying its relation to the $\\mathbb{A}^1$-Euler characteristic of quotients. As an application, we show that the compactly supported $\\mathbb{A}^1$-Euler characteristic of $\\operatorname{Sym}^n(X)$ agrees with the prediction from the power structure constructed by Pajwani-P\\\u0027al for $n = 2,3$.\n  Furthermore, we compute the compactly supported $\\mathbb{A}^1$-Euler characteristic of split toric varieties and show that the compactly supported $\\mathbb{A}^1$-Euler characteristic of all of their symmetric powers agrees with the prediction from the power structure constructed by Pajwani-P\\\u0027al.",
    "arxiv_id": "2601.05796",
    "authors": [
      "Louisa F. Br\u00f6ring"
    ],
    "categories": [
      "math.AG"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "title": "$\\mathbb{A}^1$-Euler Characteristic of Low Symmetric Powers and Split Toric Varieties",
    "url": "https://arxiv.org/abs/2601.05796",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "5529e962-a168-4558-a9c5-27fb113f2a97",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}