Mumford Linear Stability in Geometry

• Mumford Linear Stability is a numerical framework in algebraic geometry and geometric invariant theory that evaluates the stability of projective varieties and moduli spaces.
• It employs techniques such as the Hilbert–Mumford criterion and staircase one-parameter subgroups to connect algebraic invariants with analytic stability criteria.
• Its applications span moduli space construction, singularity analysis, integer programming, and signal processing, underscoring its broad impact in both theoretical and applied mathematics.

Mumford Linear Stability is a foundational concept in algebraic geometry and geometric invariant theory (GIT), introduced to interpret numerical stability criteria for projective varieties, curves, sheaves, and moduli spaces. The term encompasses various technical implementations, including the Hilbert–Mumford criterion, slope stability for nodal curves, asymptotic stability for moduli problems, and the linear behavior of algebraic invariants and homology. These approaches underpin modern moduli theory, singularity analysis, integer programming applications in commutative algebra, and the topology of moduli spaces.

1. Hilbert–Mumford Criterion and Staircase One-Parameter Subgroups

Mumford Linear Stability originates in the Hilbert–Mumford criterion, a numerical method used to determine the (semi-)stability of points in GIT quotients. For a polarized, weighted pointed nodal curve $X$, stability can be directly "read off" via the action of staircase one-parameter subgroups (1-PS) and analysis of Newton polygons (Li et al., 2011). A staircase 1-PS is a subgroup whose induced filtration on the space of sections features controlled jumps, measured by discrete invariants such as vanishing orders at nodes and marked points. The area of associated Newton polygons relates the leading coefficient of the Hilbert–Samuel polynomial to inequalities of the form

$e(J(X)) \leq E_X(p_0, p_1,\ldots)$

where $E_X$ depends linearly on the shifted weights $p_i$. The key result is that for a suitably ample polarization, a curve is Chow asymptotically stable if and only if it is slope stable, with the Hilbert–Mumford weights being positive for every nontrivial 1-PS.

2. Moduli Construction and Geometric Invariant Theory

The direct computation of Hilbert–Mumford weights via staircase 1-PS and Newton polygons is the basis for the GIT construction of moduli spaces of weighted pointed stable curves (Li et al., 2011). When embedding such curves with sufficiently high-degree line bundles, the semistability of their Hilbert scheme points (with respect to chosen linearizations) matches slope stability conditions. This framework generalizes and sharpens foundational results from Caporaso and Hassett, providing explicit criteria for compactification and moduli of singular curves. Crucially, the GIT quotient's projectivity extends even in relative situations (over a base $\operatorname{Spec}A$), and the numerical stability criterion can be computed by analyzing limits under 1-PS actions (Gulbrandsen et al., 2014).

Tool/Concept	Role in Stability Analysis	Output
Staircase 1-PS, Newton polygons	Reduce general 1-PS to controlled weight computations	Weight inequalities
Chow point, marked data	Quantify stability for weighted pointed curves	$(w(X) + H_a(X)) > 0$
Relative Hilbert–Mumford	Extend criterion to families or degenerations over a base	Projective quotients

3. K-Stability and Connections to Analytic Methods

Mumford Linear Stability also links finite-dimensional GIT criteria to analytic notions such as K-stability. For a polarized connected nodal curve $(X, \mathcal{O}_X(1))$, the Donaldson–Futaki invariant $\mathrm{DF}(X, L)$ is negative for all nontrivial test configurations if and only if the polarization is numerically proportional to the "weight polarization" $w_X$ arising from Hilbert–Mumford computations (Li et al., 2011). This establishes equivalence:

$$(X, \mathcal{O}_X(1)) \text{ is K-stable } \iff \mathcal{O}_X(1) \equiv c \cdot w_X$$

Such results demonstrate precise interactions between GIT stability (detectable via weights) and analytic stability (crucial for existence of special metrics).

4. Asymptotic and Categorical Stabilization in Moduli Theory

In the paper of moduli spaces, especially of Deligne–Mumford compactifications, linear stability concepts appear in the form of homological stabilization. The sequence of homology groups $H_i(\overline{M}_{g,n})$ is endowed with FS$^\mathrm{op}$-module structure (contravariant functors from finite sets and surjections to vector spaces), giving rise to strong asymptotic properties (Tosteson, 2018). Notably:

• The homology is finitely generated categorically, despite exponential growth in dimension.
• The generating function $\sum_n \dim(H_i(\overline{M}_{g,n})) t^n$ is rational, with the denominator’s roots corresponding to stabilization thresholds ($\{1,1/2,\ldots,1/p(g,i)\}$).
• Representation-theoretically, irreducible $S_n$-summands have Young diagrams with bounded row counts, restricting their asymptotic complexity.

5. Mumford Linear Stability for Sheaves, Coherent Systems, and Integer Programming

Generalizations of Mumford Linear Stability extend to sheaf-theoretic and commutative algebra settings. In the context of moduli spaces of stable pairs over Deligne–Mumford stacks, one defines modified Hilbert polynomial criteria mimicking Mumford's approach. Stability of pairs $(F,\varphi)$ is characterized by comparisons of normalized Hilbert polynomials, extended to stacks with ample line bundles and projective parameter spaces (Lin, 2020).

In combinatorial commutative algebra, the stability of Castelnuovo–Mumford regularity for powers of monomial ideals connects to integer programming: for $n \gg 0$, invariants like $a_i(R/\overline{I^n})$ and $\mathrm{reg}(R/\overline{I^n})$ become quasi-linear functions of $n$, i.e.,

$\mathrm{reg}(R/\overline{I^n}) = p \cdot n + e$

with explicit stability index bounds based on polyhedral combinatorics (Hoa, 2020).

6. Linear Stability in Topology, Singularities, and Metric Graphs

Mumford Linear Stability arises in the topology of moduli spaces, notably in the context of the Mumford conjecture, its b-principle form, and stabilization by concordance (Sadykov, 2021). For moduli of surfaces, rational cohomology stabilizes as genus increases, and the stable range is governed by Miller–Morita–Mumford classes:

$$H^*(\mathrm{BDiff}\,F_g;\mathbb{Q}) \cong \mathbb{Q}[\kappa_1, \kappa_2,\ldots]$$

In nonarchimedean geometry, compactified Jacobians of singular curves are constructed as Mumford models, with polyhedral decompositions of the tropical skeleton encoding linear or slope stability of sheaves (Christ et al., 2019). For semistable singularities, the Lech–Mumford constant provides an optimal value for multiplicity-colength inequalities, with field extension properties ensuring invariance under algebraic closure (Ma et al., 27 Aug 2025).

7. Extensions to Coherent Systems, Modules, and Signal Processing

Mumford Linear Stability applies to coherent systems, particularly for the paper of dual span bundles (DSB) and Butler’s conjecture (Castorena et al., 2023). Linear stability of a pair $(E,V)$ is defined by inequalities comparing degrees and dimensions of subsheaves generated by subspaces of $V$. The concept provides a weaker but more tractable criterion than slope stability of $M_{V,E}$, and is essential for explicit counterexample construction, new stabilization results, and deep connections to syzygy bundle theories.

In signal processing, Mumford–Shah models embody linear stability by preserving polynomial trends within segmented intervals. Higher-order models minimize the cost of $k$-th derivatives, ensuring piecewise polynomial solutions, and dynamic programming solvers exhibit algorithmic robustness to small perturbations—a feature labeled as Mumford Linear Stability (Storath et al., 2018).

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Mumford Linear Stability thus serves as a unifying theme across algebraic geometry, moduli theory, optimization, representation theory, and applied mathematics. It facilitates explicit criteria for stability, quantitative bounds for stabilization phenomena, and connections between finite-dimensional algebraic invariants and infinite-dimensional analytic or topological properties.