{
  "abstract": "For a $d$-dimensional Noetherian local ring $(R,\\mathfrak{m})$, with $d>0$, let $c_{\\infty}(R)$ be the limit, as $N\\to\\infty$, of the suprema of $e_R(I)/(d!\\ell_R(R/I))$ over ideals $I$ with $\\sqrt{I}=\\mathfrak{m}$ and $\\ell_R(R/I)>N$. We prove that if $R$ is complete, reduced, equidimensional, and Cohen--Macaulay on the punctured spectrum, with $d\\geq2$ and infinite residue field, then every nonzerodivisor $x\\in\\mathfrak{m}$ satisfies \\[ c_{\\infty}(R)\\leq \\max\\left\\{c_{\\infty}(R/xR), \\sup_{\\mathfrak{p}\\ne\\mathfrak{m}}e(R_{\\mathfrak{p}})\\right\\}. \\] As an application, we prove the asymptotic Lech conjecture of Huneke, Ma, Quy, and Smirnov in arbitrary characteristic: $c_{\\infty}(R)=1$ whenever $\\widehat{R}$ has isolated singularities. This also settles the conjecture of Ma and Smirnov that the Lech--Mumford constant $c_{\\mathrm{LM}}(R)$ is attained when $\\widehat{R}$ has isolated singularities and $c_{\\mathrm{LM}}(R)>1$. For reduced equidimensional Cohen--Macaulay standard graded algebras of dimension $\\geq2$ over infinite perfect fields, we prove attainment and rationality of $c_{\\mathrm{LM}}$ for the completed vertex local rings of sufficiently high Veronese subrings. In characteristic zero, a refinement of our method produces nonisolated klt affine cones in every dimension $d\\geq3$, with $d-2$-dimensional singular loci, whose completed vertex local rings have $c_{\\infty}=1$ and arbitrarily large $c_{\\mathrm{LM}}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17784",
  "author_total": 2,
  "authors": [
    {
      "name": "Hong-Ge Chen"
    },
    {
      "name": "Fei Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1p6ee1jt5k1br.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17784",
  "fulltext_url": null,
  "id": "1p6ee1jt5k1br",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17784",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7221787173",
  "pdf_url": "https://arxiv.org/pdf/2610.17784",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1p6ee1jt5k1br.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.17784",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17784"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Asymptotic Lech Bounds and Bertini Flags",
  "title_text": "Asymptotic Lech Bounds and Bertini Flags",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/1p6ee1jt5k1br",
  "venue": "arXiv",
  "versions": []
}