Integer-Constrained Cone Singularities

• Integer-constrained cone singularities are defined by discrete, integer-valued invariants such as isotropy orders, multiplicities, and cone angles, ensuring rigidity and boundedness.
• They appear in diverse settings including toric geometry, minimal hypersurfaces, and discrete conformal mapping, each employing specific integer constraints to control local behavior.
• Key techniques involve reduction to log Fano boundedness, discrete Laplacian solvers, and mixed-integer optimization, enabling efficient classification and computational applications.

Integer-constrained cone singularities arise in algebraic geometry, geometric analysis, combinatorics, and computational geometry as local models for singularities where discrete, typically integer-valued, invariants constrain the local analytic, geometric, or combinatorial data. Across distinct settings, these singularities are characterized by the appearance of integers as isotropy orders, multiplicities, cone angles, or other discrete parameters, which enforce finiteness, boundedness, and rigidity properties in classification and applications.

1. Algebraic and Geometric Definitions

Integer-constrained cone singularities appear in several precise mathematical frameworks:

• Toric and Log Calabi–Yau Geometry: A cone singularity on a normal affine variety $X$ is defined via an effective $\mathbb{G}_m$-action with a unique fixed point $x$ in $X$ lying in the closure of every orbit. The isotropy order at any $y \neq x$ is $|{\rm Stab}_{k^*}(y)| \in \mathbb{Z}_{>0}$, forcing isotropy to be integer-valued and often uniformly bounded above by a fixed $N$ (Moraga, 2018).
• Discrete and Combinatorial Constructions: On discrete surfaces or planar graphs, integer cone singularities are encoded in the assembly of Euclidean rectangles (or more general polygons) at each vertex, resulting in cone angles that are integer multiples of $2\pi$ (or, in refined versions, of $\pi/2$ or $\pi$), imposing rigid combinatorial constraints (Hersonsky, 2015, Hersonsky, 2010).
• Geometric Measure Theory: For immersed minimal hypersurfaces, tangent cones at isolated singularities decompose with integer multiplicity $q \geq 1$ into regular stationary cones. This integer multiplicity encodes both density and sheeting structure in the limiting varifold (Edelen et al., 27 Jan 2024).
• Conformal and Computational Geometry: On triangulated meshes, cone singularities at vertices are specified by an integer multiple of a fundamental angle, under topological constraints (e.g., the discrete Gauss–Bonnet theorem), and subject to optimization for geometric objectives like minimal area distortion (Du et al., 24 Dec 2025).
• Analytic and Topological Invariants: For complex curve singularities, integer data—auxiliary multiplicities—are extracted from Puiseux expansions, governing the stratification of higher-order tangent cones and bi-Lipschitz classification (Flores et al., 2021).

2. Structural Theorems and Boundedness

A foundational property in the theory of integer-constrained cone singularities is boundedness, which may manifest in the number of deformation types, values taken by invariants, or topological types.

• Boundedness of $\epsilon$-log-canonical Cone Singularities: For dimension $d$, fixed $\epsilon>0$, and isotropy order at most $N$, the class $C_{d, \epsilon, N}$ of $d$-dimensional $\epsilon$-log-canonical cone singularities with isotropies bounded by $N$ is a bounded family. Explicitly,

$$\Bigl\{\,X : \dim X=d,\, X\text{ is } \epsilon\text{-lc},\; \text{isotropy}(y)\leq N\,\forall y\,\Bigr\}$$

is bounded (Moraga, 2018). This leads to finiteness for minimal log discrepancies and for algebraic fundamental group orders.

• Discrete Uniformization and Gauss–Bonnet Constraints: For rectangularly tiled flat surfaces from discrete harmonic data, all cone angles at vertices are $2\pi k$ for $k\in\mathbb{N}$, and the sum of cone orders satisfies $\sum k_v = 2-2g$ (genus $g$). This guarantees uniqueness up to isometry and reflects rigid combinatorial constraints (Hersonsky, 2015, Hersonsky, 2010).
• Tangent Cones with Integer Multiplicity: For immersed minimal hypersurfaces, uniqueness and regularity theorems hold for tangent cones with any prescribed integer multiplicity, and the covering structure is rigidly finite for each integer $q$ (Edelen et al., 27 Jan 2024).

3. Integer Constraints in Discrete and Computational Settings

Discrete geometric constructions often require both local and global integer constraints for correctness and optimality.

• Rectangular Tiling and Integer Cone Angles: In energy-preserving tilings derived from solutions to (mixed) Dirichlet-Neumann problems on planar graphs, the local angle at each vertex is a sum of right angles, and the resulting deficit (curvature) is always an integer multiple of $2\pi$. This is forced globally by the discrete Gauss-Bonnet relation, ensuring that only integer cone singularities arise in the discrete uniformization map to flat surfaces (Hersonsky, 2015, Hersonsky, 2010).
• Efficient Optimization for Conformal Parameterization: In computational geometry, sparse integer-constrained cone configurations (positions and angles) are determined via alternating minimization—solving quadratic programs over discrete variables constrained by both local integer angle requirements and a global sum constraint, typically enforced by the Euler characteristic (Du et al., 24 Dec 2025). Integer holonomy constraints along non-contractible loops generalize these relations in higher genus.

4. Higher-Order Cones, Multiplicities, and Classification

Integer invariants dominate classification and invariant theory for cone singularities in analytic and geometric settings.

• Auxiliary Multiplicities and Whitney Cones: For the fifth Whitney cone $C_5(X,0)$ of a plane curve germ $(X,0)$, auxiliary multiplicities extracted from Puiseux parametrizations index the collection of two-planes comprising $C_5(X,0)$. These integers classify bi-Lipschitz types and yield sharp upper bounds on the number of irreducible components of $C_5$. Their arithmetic, including gcds for tangent branches and divisor chains for irreducible branches, controls geometric properties and moduli (Flores et al., 2021).
• Moduli and Toroidal Compactifications: In the perfect cone decomposition for moduli of abelian varieties, integral generators of the cones (in the lattice of quadratic forms) yield basicness criteria: a cone is basic if its primitive integral generators form a lattice basis (index 1), so the integer structure of the generators induces finiteness theorems for singular loci and controls local singularity types via simpliciality and basicness (Sikirić et al., 2013).

5. Proof Techniques and Algorithmic Frameworks

A range of proof architectures integrate integer-constrained structures:

• Reduction to Log Fano Boundedness: In the boundedness theorem, every cone singularity is encoded via a pair $(Y,D)$. Bounded isotropy implies bounded denominator in $D = \sum (p_i/q_i)D_i$, thus reducing the problem to the boundedness of log Fano pairs with coefficients in a finite set, leveraging Birkar's theorem for strictly log-bounded families (Moraga, 2018).
• Discrete Laplacian Solvers and Index Lemma: The uniqueness and existence of rectangle-tiled metrics with integer cone singularities are established through discrete potential theory, Laplacian linear systems, and combinatorial index arguments reflecting Euler characteristic constraints (Hersonsky, 2015, Hersonsky, 2010).
• Sheeting, Łojasiewicz–Simon Inequalities, and Covering Arguments: Regularity and uniqueness results for minimal hypersurfaces with integer-multiplicity tangent cones build on sharp sheeting theorems and quantifiable decay rates on $q$-fold covers, reducing infinite possibilities to finite topological types due to the integer covering degree (Edelen et al., 27 Jan 2024).
• Mixed-Integer Optimization: Efficient computational methods for placing and optimizing integer-constrained cones rely on reducing the problem size to the cardinality of the cones, leveraging MIQP (mixed-integer quadratic programming) and exploiting sparse updates through active variable selection (Du et al., 24 Dec 2025).

6. Examples, Applications, and Open Problems

Integer-constrained cone singularities underpin both structure theory and explicit classification:

• In dimension $d=2$ and isotropy bound $N=2$, the classification reduces to a finite list involving cyclic quotient singularities of type $\frac{1}{2}(1,p)$ with $p$ odd and constrained by discrepancy computations (Moraga, 2018).
• For rectangle-tiled flat surfaces on triangulated genus $g$ surfaces, only $2\pi k$ cone angles are possible, and sum constraints enforce finiteness in the configuration space of singularities (Hersonsky, 2015).
• Application to Fano cones: Affine Fano cones with bounded weight denominators $q_i \leq N$ form finitely many deformation families, relevant for K-stability considerations (Moraga, 2018).
• In computational conformal mapping, the integer-constrained framework enables rotationally seamless, low-distortion parameterizations with provable speedups and optimality under discrete constraints (Du et al., 24 Dec 2025).
• For complex curve singularities, the list of auxiliary multiplicities determines the Lipschitz equivalence class, bound the number of components of $C_5$ by explicit arithmetic in multiplicities, and exhibit rigidity under equisingular deformations. Notably, the number of irreducible components of the fifth Whitney cone may jump even in bi-Lipschitz equisingular families (Flores et al., 2021).

Open problems include the enumeration of higher-dimensional Gorenstein cone singularities with bounded isotropy, the structural behavior when only partial integer constraints are enforced, and the stability of invariants under varying bounds or relaxed log canonicity, as well as explicit algorithmic classification in computational applications (Moraga, 2018).