Three-Parameter Pre-Stability Conditions

• The topic defines pre-stability conditions using three parameters that establish regions of well-posedness, mass conservation, and positivity across diverse mathematical frameworks.
• Methodologies employ central charge constructions, tilting procedures, and inf-sup conditions to transition from abstract stability notions to practical analytical and numerical criteria.
• Applications span Bridgeland stability in derived categories, coherent systems on curves, lattice Boltzmann schemes, ERGMs, and PDE variational formulations like Biot’s consolidation.

A three-parameter family of pre-stability conditions arises in several mathematical and physical contexts, notably in the paper of derived categories in algebraic geometry, lattice Boltzmann methods in numerical analysis, variational formulations of PDE models such as Biot’s consolidation, and exponential random graph models. Each framework formulates a notion of stability or “pre-stability” using three fundamental real or structural parameters, charting a structured region in parameter space where objects or solutions satisfy well-posedness, mass conservation, positivity, or other physically or algebraically significant constraints.

1. Bridgeland-Type Three-Parameter Pre-Stability in Algebraic Geometry

Central to the modern derived category approach for smooth projective varieties and their products is a systematic method to construct stability conditions parameterized by three independent real parameters. For $X$ a smooth projective variety of dimension $n$ and $C$ a smooth projective curve, a base stability condition $(Z_X, \A_X)$ is assumed on $D^b(X)$, with $Z_X: K(D^b(X))\to\CC$ and $\A_X$ Noetherian. The heart $\A_C$ of a bounded t-structure on $D^b(X\times C)$ is defined via the Abramovich–Polishchuk construction, considering objects $E$ such that pushforwards $p_*(E\otimes q^*\OO_C(n))$ lie in $\A_X$ for large $n$.

On this heart, the pre-stability construction is governed by three parameters:

• $B_X \in \NS(X)_\RR$ (numerical class shift)
• $\omega_X \in \Amp(X)_\RR$ (polarization on $X$)
• $\omega_C > 0$ (polarization on $C$)

A polynomial central charge is given by integrating the twisted Chern character against the exponential of the complexified Kähler form,

$$Z_{(B_X,\omega_X,\omega_C)}(E) = -\int_{X\times C} e^{-i(\omega_X \oplus \omega_C)} \cup \ch^{B_X}(E),$$

where $\ch^{B_X}(E) = e^{-B_X}\ch(E)$. The tilt process, based on leading-coefficient stability ($\mu_2$), yields a new heart $\A_{(B_X,\omega_X,\omega_C)}$. The combined procedure yields a continuous family of pre-stability conditions, and, under suitable positivity and support properties (as shown via the Bayer–Macrì positivity lemma and subsequent works), a genuine Bridgeland stability condition on $D^b(X\times C)$ (Liu, 2019).

2. Three-Parameter Family for Coherent Systems on Integral Curves

In the context of coherent systems on an integral projective curve $C$, the category $\A = \CohSys(C)$ consists of tuples $(E,V,\varphi)$ with $E$ a coherent sheaf and $V$ a subspace of $H^0(E)$. The objects are classified by their type vector $v(\cE) = (n,d,k) \in \Z^3$, corresponding to rank, degree, and subspace dimension.

A sequence of tilts is applied:

• The first two tilting parameters, $\alpha \ge 0$ and $\beta \in \RR$, are used to define a heart $\A_{(\alpha,\beta)}$ via the torsion pair associated to the slope function $\mu_\alpha(\cE) = \frac{d}{n} + \alpha \frac{k}{n}$.
• A third parameter $\gamma > 1$ is introduced in the central charge

$$Z_\alpha^{\beta,\gamma}(n,d,k) = (d + \gamma n - k) + i (d + \alpha k - \beta n).$$

Within a region $\mathsf S \subset \RR_{\ge 0} \times \RR_{\ge 0} \times \RR_{> 1}$, this setup defines a pre-stability condition; when the support property is verified, this yields a genuine Bridgeland stability condition of complex dimension 3 (Jardim et al., 16 Nov 2025). The wall/chamber structure for stability involves explicit expressions for destabilizing “walls” as functions of the parameters, with detailed semistability criteria derived for torsion, free, and complete tilted systems.

3. Pre-Stability Regions in Lattice Boltzmann Three-Velocities Schemes

In the lattice Boltzmann framework, a three-parameter family of pre-stability conditions naturally emerges in the D1Q3 relaxation schemes with relative velocity $u$ and relaxation parameters $\omega_1, \omega_2$. The global non-negativity of the post-collision distribution—a pre-stability property equivalent to preservation of the maximum principle—is characterized by the constraints: $$\max(\omega_2 - 1, |\bar u|) \leq 2\gamma \leq \min(2 - \omega_1 - |\bar u - \omega_1 V|,\, \omega_1 - |\bar u + \omega_1 V|,\, \omega_2 - |\omega_1 V|),$$ with $\bar u = 2u(\omega_1 - \omega_2)$ and $$\gamma = \frac{\omega_2}{6}(1-\alpha) - u(\omega_1 - \omega_2)V$$. This defines a convex polyhedral region $\mathcal{R} \subset \RR^3$ in the coordinate space $(u, \omega_1, \omega_2)$, where the pre-stability (non-negativity) property holds (Dubois et al., 2019).

4. Three-Parameter Pre-Stability in Exponential Random Graph Models

For exponential random graph models (ERGMs) with three sufficient statistics, the region of parameters $\theta = (\theta_1,\theta_2,\theta_3)$ making a target graph $G^*$ locally stable with respect to a set of alternatives is given by

$C = \{\theta \in \RR^3 : \theta \cdot \Delta^{(i)} > 0 \quad \forall i=1,...,m\},$

where each $\Delta^{(i)}$ is the vector of the change-scores for the statistics between $G^*$ and $G^{(i)}$ (Yu et al., 2019). This intersection of half-spaces is a convex cone, and the boundaries correspond to critical changes in stability. This formalizes the “pre-stability region” in ERGMs as the collection of parameter values for which the target structure is strictly locally favored.

5. Three-Parameter Stability in PDE Variational Formulations

In the analysis of Biot’s consolidation model, the PDE system for displacement $u$, Darcy flux $v$, and pore pressure $p$ admits a three-parameter family of parameter-dependent norms:

• Displacement: $$\|u\|_{U}^2 = (\epsilon(u),\epsilon(u)) + \lambda\|\operatorname{div} u\|^2$$
• Flux: $$\|v\|_V^2 = R_p^{-1}\|v\|^2 + \gamma^{-1}\|\operatorname{div} v\|^2$$
• Pressure: $\|p\|_P^2 = \gamma\|p\|^2$

Here, $\lambda$ (Lamé parameter), $R_p^{-1}$ (inverse poroelastic modulus), and $\gamma$ (dependent on all model parameters) serve as the three parameters. Bilinear form continuity and inf–sup stability are obtained with constants independent of these parameters, defining a three-parameter “pre-stability” region for which the variational problem is uniformly stable (Hong et al., 2017).

6. Structural and Geometric Properties of Pre-Stability Regions

Across these frameworks, the region of admissible parameters is either a convex polyhedron (as in the lattice Boltzmann and ERGM cases) or an open subset defined by explicit inequalities (as for stability conditions in derived categories). The faces and intersection structure of these regions correspond to wall-crossing loci, criticality in parameter space, or phase transitions. In the algebraic geometry context, the parameter domain may be identified with a real manifold of stability conditions.

Framework	Parameters	Pre-Stability Region Structure
Bridgeland (Product Varieties)	$B_X, \omega_X, \omega_C$	Continuous family, wall/chamber structure
Coherent Systems on Curves	$\alpha, \beta, \gamma$	Explicit region $\mathsf S$, real 3-manifold
Lattice Boltzmann (D1Q3)	$u, \omega_1, \omega_2$	Convex polyhedron, piecewise planar faces
ERGM Local Stability	$\theta_1, \theta_2, \theta_3$	Convex cone in $\RR^3$
Biot Model Variational Stability	$\lambda, R_p^{-1}, \gamma$	Uniform inf-sup region, parameter-robust

Table 1: Three-parameter pre-stability regions across frameworks.

7. Significance and Applications

The introduction and characterization of three-parameter families of pre-stability conditions provide the infrastructure for analyzing wall-crossing phenomena, constructing moduli spaces, establishing robust numerical schemes, and proving parameter-uniform well-posedness in applied models. In algebraic geometry, this underpins the existence and structure of complex manifolds of stability conditions; in numerical PDEs, it guarantees robust preconditioning and optimal discretization; in probabilistic models, it gives precise control over stability of network structures.

These results highlight the unifying mathematical theme that stability—expressed as a set of inequalities or inf-sup conditions—naturally organizes parameter space into well-understood geometric regions, enabling both theoretical analysis and practical computation across a wide spectrum of disciplines.