Lech–Mumford Constant in Local Rings

• The Lech–Mumford constant is an invariant of Noetherian local rings defined as the supremum of the normalized ratio between Hilbert–Samuel multiplicity and colength.
• It connects classical multiplicity inequalities to modern stability notions and is crucial in identifying log canonical and semi-log canonical singularities.
• Computational techniques leverage reductions to integrally closed and monomial ideals, yielding sharp asymptotic bounds and effective multiplicity estimates.

The Lech–Mumford constant of a Noetherian local ring $(R,\mathfrak m)$ of dimension $d$ is the invariant

$c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$

where $e(I)$ is the Hilbert–Samuel multiplicity of $I$ and $\lambda(-)$ denotes length. In this normalization, the invariant is the optimal constant in Lech’s inequality after dividing multiplicity by $d!$ times colength, and it does not include factors of $e(R)=e(\mathfrak m)$. The modern theory of $c_{LM}(R)$ connects this optimal multiplicity–colength ratio to stability notions for local rings and, under mild hypotheses, to semi-log canonical and log canonical singularities (Ma et al., 27 Aug 2025).

1. Definition, normalization, and relation to Lech-type inequalities

The Hilbert–Samuel multiplicity admits the formula

$$e(I)=d!\,\lim_{n\to\infty}\frac{\lambda(R/I^n)}{n^d}.$$

The classical Lech inequality states that for any $d$0-primary ideal $d$1 in a Noetherian local ring $d$2 of dimension $d$3,

$d$4

Lech observed that the inequality is never sharp when $d$5. The invariant $d$6 optimizes this inequality in the normalization above: one always has

$d$7

where $d$8 is the unmixed quotient and $d$9 is the largest submodule of dimension $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$0. In particular,

$c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$1

if and only if either $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$2, or $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$3 and $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$4; this is the uniform Lech theorem (Ma et al., 27 Aug 2025).

A broader packaging of related optimal constants appears in "A generalization of an inequality of Lech relating multiplicity and colength" (Huneke et al., 2017). In that framework, the colength constant $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$5 is the smallest number such that

$c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$6

for all $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$7-primary $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$8, and Lech’s inequality gives $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$9. The same source also packages mixed, product, and generator versions of Lech-type bounds as “Lech–Mumford constants.” This suggests that $e(I)$0 is the normalized local-ring invariant underlying a wider landscape of optimal multiplicity inequalities.

2. Structural properties and functorial behavior

The invariant admits several reductions that make it computable in practice. One may restrict the supremum to $e(I)$1-primary integrally closed ideals, since passing to $e(I)$2 preserves multiplicity and reduces colength. It is invariant under completion and under passage to the unmixed quotient: $e(I)$3 If $e(I)$4 is a quotient of $e(I)$5 with $e(I)$6, then

$e(I)$7

For $e(I)$8, one has $e(I)$9 (Ma et al., 27 Aug 2025).

The invariant behaves well in multigraded and monomial settings. If $I$0 is multigraded over a local base, then $I$1 may be computed by homogeneous $I$2-primary ideals; in particular,

$I$3

and $I$4 can be computed by homogeneous ideals $I$5. For $I$6 with $I$7 a polynomial ring over a field, weights or term orders give

$I$8

and similarly for Gröbner degenerations with a monomial order.

Slicing and localization impose monotonicity constraints. If $I$9 is a parameter element, then

$\lambda(-)$0

In particular,

$\lambda(-)$1

For $\lambda(-)$2, equality $\lambda(-)$3 forces that the supremum in $\lambda(-)$4 is not attained. If $\lambda(-)$5 with $\lambda(-)$6, then

$\lambda(-)$7

and

$\lambda(-)$8

Under equidimensionality and catenarity, lim-stability localizes: if $\lambda(-)$9 is lim-stable, then $d!$0 is lim-stable for all $d!$1.

Flat and finite maps also admit comparison inequalities. If $d!$2 is flat local with $d!$3, then

$d!$4

If $d!$5 is finite over a domain $d!$6, then

$d!$7

In particular, finite birational extensions satisfy $d!$8.

In families, the established result is a weak semicontinuity statement. If $d!$9 is finite type flat with a section $e(R)=e(\mathfrak m)$0, and $e(R)=e(\mathfrak m)$1, then

$e(R)=e(\mathfrak m)$2

Along a DVR base, $e(R)=e(\mathfrak m)$3 at the special fiber dominates generically. The authors conjecture upper semicontinuity in families.

3. Stability hierarchy attached to $e(R)=e(\mathfrak m)$4

The asymptotic version of the invariant is

$e(R)=e(\mathfrak m)$5

and this limit exists and is $e(R)=e(\mathfrak m)$6 (Ma et al., 27 Aug 2025).

Notion	Condition	Immediate relation
Lech-stable	$e(R)=e(\mathfrak m)$7	implies semistable
Semistable	$e(R)=e(\mathfrak m)$8	implies lim-stable
Lim-stable	$e(R)=e(\mathfrak m)$9	reducedness consequence
Stable	semistable and the supremum in $c_{LM}(R)$0 is not attained	implied by Lech-stable for $c_{LM}(R)$1

The formal implications are

$c_{LM}(R)$2

and for $c_{LM}(R)$3,

$c_{LM}(R)$4

A lim-stable ring, and more generally a semistable Cohen–Macaulay ring, has $c_{LM}(R)$5 reduced; in particular, lim-stable implies reduced.

The low-dimensional classifications are explicit. In dimension $c_{LM}(R)$6, $c_{LM}(R)$7 if and only if $c_{LM}(R)$8 is a field; semistability and lim-stability are equivalent to being a field; and $c_{LM}(R)$9 is never stable, since $$e(I)=d!\,\lim_{n\to\infty}\frac{\lambda(R/I^n)}{n^d}.$$0 is $$e(I)=d!\,\lim_{n\to\infty}\frac{\lambda(R/I^n)}{n^d}.$$1-dimensional and the supremum is attained by the maximal ideal. In dimension $$e(I)=d!\,\lim_{n\to\infty}\frac{\lambda(R/I^n)}{n^d}.$$2, $$e(I)=d!\,\lim_{n\to\infty}\frac{\lambda(R/I^n)}{n^d}.$$3 always. Moreover:

• Lech-stable $$e(I)=d!\,\lim_{n\to\infty}\frac{\lambda(R/I^n)}{n^d}.$$4 stable $$e(I)=d!\,\lim_{n\to\infty}\frac{\lambda(R/I^n)}{n^d}.$$5 the unmixed part of $$e(I)=d!\,\lim_{n\to\infty}\frac{\lambda(R/I^n)}{n^d}.$$6 is regular.
• Semistable $$e(I)=d!\,\lim_{n\to\infty}\frac{\lambda(R/I^n)}{n^d}.$$7 lim-stable $$e(I)=d!\,\lim_{n\to\infty}\frac{\lambda(R/I^n)}{n^d}.$$8 the unmixed part of $$e(I)=d!\,\lim_{n\to\infty}\frac{\lambda(R/I^n)}{n^d}.$$9 is either regular or a node (double normal crossing).

The terminology is therefore hierarchical rather than synonymous. Simple normal crossings already separate the notions: for

$d$00

one has Lech-stable for $d$01, semistable but not stable for $d$02, and stable but not Lech-stable for $d$03. In particular,

$d$04

4. Links with semi-log canonical, log canonical, and canonical singularities

The main structural advance of the theory is a direct connection between $d$05-stability and singularities from the minimal model program. Let $d$06 be essentially of finite type over a field of characteristic $d$07, satisfying Serre’s $d$08 and $d$09, and $d$10-Gorenstein. If $d$11 is semistable, equivalently $d$12, then $d$13 is semi-log canonical. This is Main Theorem A (Ma et al., 27 Aug 2025).

For normal rings, the lim-stable condition is stronger. Let $d$14 be an excellent normal local domain admitting a dualizing complex. Assume either $d$15, or $d$16 is essentially of finite type over a field of characteristic $d$17 and numerically $d$18-Gorenstein. Then: $d$19

$d$20

These are the two parts of Main Theorem B. More generally, beyond $d$21-Gorenstein hypotheses, under $d$22, $d$23, $d$24-Gorenstein, and either $d$25 or “essentially finite type over a field of characteristic $d$26,” lim-stability implies semi-log canonicity.

The proof strategy combines birational geometry with asymptotic multiplicity theory. One uses a log canonical or semi-log canonical modification $d$27, available under the stated conditions by the Odaka–Xu theorem and Hashizume’s numerically $d$28-Gorenstein extension. From the exceptional geometry one constructs a graded family of $d$29-primary ideals via pullbacks and anti-nef divisors, and then evaluates the asymptotic behavior of lengths and multiplicities using asymptotic Riemann–Roch on a projective birational model. A key inequality compares the second asymptotic coefficient of colengths to discrepancies; negativity coming from non-log canonical centers forces $d$30 to be strictly $d$31, contradicting lim-stability. The technical bridge is a derivative criterion: if $d$32 for the Hilbert–Poincaré numerator $d$33, then

$d$34

5. Asymptotic and computational apparatus

For a Noetherian graded family $d$35 with $d$36 and $d$37, the Hilbert series

$d$38

is rational with denominator $d$39: $d$40 for a rational $d$41 with poles of order $d$42. If $d$43, then for any $d$44 one has the lower bound

$d$45

provided

$d$46

This is the quantitative tool used to force $d$47 from suitable asymptotics (Ma et al., 27 Aug 2025).

Rational powers and Rees valuations give a refined supply of integrally closed ideals. For an ideal $d$48 and rational $d$49, define

$d$50

These rational powers are integrally closed and well-defined via the Rees valuations $d$51 of $d$52; the Rees period

$d$53

is a common denominator, and

$d$54

In analytically unramified rings, the functions

$d$55

are eventually quasi-polynomials with leading term $d$56. The second-order term is computed via asymptotic Riemann–Roch: $d$57 where $d$58 on $d$59.

The asymptotic Riemann–Roch statements used in this theory are also explicit. If $d$60 is projective over an Artinian base, equidimensional and $d$61, and $d$62 is a line bundle, then

$d$63

In the $d$64 setting with Weil divisors $d$65 principal in codimension one,

$d$66

These formulas identify the second asymptotic coefficient that enters the derivative criterion.

On the computational side, several reductions are available. One has

$d$67

so associated graded rings reduce the problem to multigraded or monomial settings. Monomial and initial ideals preserve or improve $d$68, and inclusion–exclusion schemes for lengths control sums and intersections of coordinate ideals. Mixed multiplicities, through Risler–Teissier theory, express mixed multiplicities in terms of reductions and intersection numbers and are used in improving three-dimensional bounds.

Low-dimensional sharp inequalities are particularly important. For $d$69 and monomial $d$70-primary $d$71, Mumford’s sharp inequality is

$d$72

For $d$73 and any $d$74-primary ideal $d$75,

$d$76

with equality for powers of the maximal ideal. These formulas are the effective bounds behind many model computations and semistability arguments.

6. Model classes, explicit values, and open directions

The theory includes a substantial list of explicit examples and counterexamples (Ma et al., 27 Aug 2025).

Class	Outcome	Note
Simple normal crossings $d$77	$d$78 Lech-stable; $d$79 semistable not stable; $d$80 stable not Lech-stable	$d$81
Determinantal rings	Lech-stable	generic determinantal hypersurface and maximal minors
Veronese $d$82	changes with $d$83	thresholds at $d$84
Polygonal cones	semistability up to $d$85	stability up to $d$86 in the stated families

For determinantal rings, the generic determinantal hypersurface $d$87 is Lech-stable because it degenerates to simple normal crossings. More generally, if $d$88 is the ideal of generic $d$89 minors of an $d$90 matrix with $d$91, then $d$92 is Lech-stable.

In dimension $d$93, pseudo-rational normal local rings exhibit a precise relation between $d$94 and the multiplicity. If $d$95 is pseudo-rational normal of dimension $d$96, then for any integrally closed $d$97-primary $d$98, stability of $d$99 implies

$c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$00

and in particular $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$01 if $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$02 is not regular. For Gorenstein rational double points (ADE), $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$03 is Lech-stable.

Minimally elliptic normal surface singularities also admit exact values. If $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$04 is minimally elliptic of degree $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$05, with fundamental cycle $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$06 and irreducible exceptional divisor, then

$c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$07

The constant is attained by the maximal ideal for $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$08, by $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$09 for $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$10, and in degree $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$11 by a deeper integrally closed ideal.

For cones and Veronese subrings, semistability occurs in bounded ranges. Elliptic polygonal $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$12-cones in $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$13 are semistable if and only if $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$14; the case $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$15 is stable, while $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$16 is semistable but not stable. Rational polygonal $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$17-cones in $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$18 are semistable if and only if $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$19, and stable if and only if $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$20. For the completed two-variable Veronese subrings,

$c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$21

The semistability results also cover hypersurface surface singularities. Most two-dimensional semi-log canonical hypersurface singularities are semistable, including the families $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$22, degenerate cusps, and the simple normal crossings cases. For the simple elliptic families $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$23, $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$24, and $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$25, semistability is conjectured; degeneration methods are insufficient because initial terms do not capture elliptic geometry.

Several open problems remain central. One concerns asymptotic attainment: if $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$26 has isolated singularity and $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$27, then the conjecture predicts that the supremum in $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$28 is attained, hence $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$29. This is proved in positive characteristic for perfect residue fields and extends to two-dimensional Gorenstein normals, but is unknown beyond that setting. A second problem is upper semicontinuity in families. A third is the “Best Lech” conjecture, which proposes an optimal higher-dimensional Lech inequality for monomial ideals using Stirling-number coefficients; it implies the HSV conjecture on refinements of Lech’s inequality, and the cases $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$30 are proved. A fourth is that all two-dimensional semi-log canonical hypersurfaces, including the elliptic families, should be semistable.

The practical consequences are correspondingly concrete. Computing $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$31 reduces to integrally closed ideals and often further to homogeneous or monomial degenerations; sharp two- and three-dimensional Lech-type inequalities, inclusion–exclusion, and asymptotic graded-family methods then become effective. Stability is compatible with completion, unmixed reduction, flat base change, and suitable localization. In Cohen–Macaulay strict complete intersections with $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$32 a complete intersection of degrees $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$33, lim-stability forces $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$34, where $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$35 is the embedding dimension, while Lech-stability forces a strict inequality. Large multiplicity precludes lim-stability: for dimension $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$36, there is $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$37 with

$c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$38

for example $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$39 and $c_{LM}(R)\coloneqq \sup_{\sqrt{I}=\mathfrak m}\left\{\frac{e(I)}{d!\,\lambda(R/I)}\right\} =\sup_{I\text{ %%%%0%%%%-primary}}\frac{e(I)}{d!\,\lambda(R/I)},$40. These results position the Lech–Mumford constant as both a sharp multiplicity invariant and a stability detector for singularities.