Asymptotic Lech Bounds and Bertini Flags

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 \(\cLM\).