Asymptotic Lech Bounds and Bertini Flags
Abstract
For a d-dimensional Noetherian local ring (R,m), with d>0, let c∞(R) be the limit, as N→∞, of the suprema of eR(I)/(d!ℓR(R/I)) over ideals I with √I=m and ℓR(R/I)>N. We prove that if R is complete, reduced, equidimensional, and Cohen--Macaulay on the punctured spectrum, with d≥2 and infinite residue field, then every nonzerodivisor x∈m satisfies c∞(R)≤ max{c∞(R/xR), supp≠me(Rp)}. As an application, we prove the asymptotic Lech conjecture of Huneke, Ma, Quy, and Smirnov in arbitrary characteristic: c∞(R)=1 whenever R̂ has isolated singularities. This also settles the conjecture of Ma and Smirnov that the Lech--Mumford constant cLM(R) is attained when R̂ has isolated singularities and cLM(R)>1. For reduced equidimensional Cohen--Macaulay standard graded algebras of dimension ≥2 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≥3, with d−2-dimensional singular loci, whose completed vertex local rings have c∞=1 and arbitrarily large cLM.
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BibTeX
@misc{chen2026asymptotic,
title = {{Asymptotic Lech Bounds and Bertini Flags}},
author = {Hong-Ge Chen and Fei Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17784},
eprint = {2610.17784},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17784},
}
RIS
TY - UNPB
TI - Asymptotic Lech Bounds and Bertini Flags
AU - Hong-Ge Chen
AU - Fei Liu
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.17784
UR - https://arxiv.org/abs/2610.17784
AB - For a d-dimensional Noetherian local ring (R,m), with d>0, let c_∞(R) be the limit, as N→∞, of the suprema of e_R(I)/(d!ℓ_R(R/I)) over ideals I with √I=m and ℓ_R(R/I)>N. We prove that if R is complete, reduced, equidimensional, and Cohen--Macaulay on the punctured spectrum, with d≥2 and infinite residue field, then every nonzerodivisor x∈m satisfies c_∞(R)≤ max{c_∞(R/xR), sup_p≠me(R_p)}. As an application, we prove the asymptotic Lech conjecture of Huneke, Ma, Quy, and Smirnov in arbitrary characteristic: c_∞(R)=1 whenever R̂ has isolated singularities. This also settles the conjecture of Ma and Smirnov that the Lech--Mumford constant c_LM(R) is attained when R̂ has isolated singularities and c_LM(R)>1. For reduced equidimensional Cohen--Macaulay standard graded algebras of dimension ≥2 over infinite perfect fields, we prove attainment and rationality of c_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≥3, with d−2-dimensional singular loci, whose completed vertex local rings have c_∞=1 and arbitrarily large c_LM.
ER -