Asymptotic Lech Bounds and Bertini Flags

For a $d$-dimensional Noetherian local ring $(R,\text{m})$, with $d>0$, let $\text{cinf}(R)$ be the limit, as $N\to \mathrm{\infty }$, of the suprema of $e_R(I)\mathrm{/}(d!{\text{length}}_R(R\mathrm{/}I))$ over ideals $I$ with $\sqrt{I}=\text{m}$ and ${\text{length}}_R(R\mathrm{/}I)>N$. We prove that if $R$ is complete, reduced, equidimensional, and Cohen–Macaulay on the punctured spectrum, with $d\ge 2$ and infinite residue field, then every nonzerodivisor $x\in \text{m}$ satisfies $$\text{cinf}(R)\le \max \{\text{cinf}(R\mathrm{/}xR),\underset{\text{fp}\mathrm{\ne }\text{m}}{\sup }e(R_{\text{fp}})\}\mathrm{.}$$ As an application, we prove the asymptotic Lech conjecture of Huneke, Ma, Quy, and Smirnov in arbitrary characteristic: $\text{cinf}(R)=1$ whenever $\widehat{R}$ has isolated singularities. This also settles the conjecture of Ma and Smirnov that the Lech–Mumford constant $\text{cLM}(R)$ is attained when $\widehat{R}$ has isolated singularities and $\text{cLM}(R)>1$. For reduced equidimensional Cohen–Macaulay standard graded algebras of dimension $\ge 2$ over infinite perfect fields, we prove attainment and rationality of $\text{cLM}$ 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 $d\ge 3$, with $d-2$-dimensional singular loci, whose completed vertex local rings have $\text{cinf}=1$ and arbitrarily large $\text{cLM}$.