Asymptotic Lech Bounds and Bertini Flags

Hong-Ge Chen; Fei Liu

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arXiv · 2610.17784

Asymptotic Lech Bounds and Bertini Flags

Hong-Ge Chen·Fei Liu

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Abstract

For a $d$-dimensional Noetherian local ring $(R,\m)$, with $d>0$, let $\cinf(R)$ be the limit, as $N\to\infty$, of the suprema of $e_R(I)/(d!\length_R(R/I))$ over ideals $I$ with $\sqrt I=\m$ and $\length_R(R/I)>N$. 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 $x\in\m$ satisfies \[ \cinf(R)\leq \max\left\{\cinf(R/xR),\sup_{\fp\ne\m}e(R_{\fp})\right\}. \] As an application, we prove the asymptotic Lech conjecture of Huneke, Ma, Quy, and Smirnov in arbitrary characteristic: $\cinf(R)=1$ whenever $\widehat R$ has isolated singularities. This also settles the conjecture of Ma and Smirnov that the Lech--Mumford constant $\cLM(R)$ is attained when $\widehat R$ has isolated singularities and $\cLM(R)>1$. For reduced equidimensional Cohen--Macaulay standard graded algebras of dimension $\geq2$ over infinite perfect fields, we prove attainment and rationality of $\cLM$ 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 $\cinf=1$ and arbitrarily large $\cLM$.

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BibTeX RIS

Hong-Ge Chen, Fei Liu. 2026-10-13. Asymptotic Lech Bounds and Bertini Flags. https://arxiv.org/abs/2610.17784

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