Ideal Point Gluing of Curves

• Ideal point gluing of curves is a process that joins algebraic or geometric curves at prescribed points to create new families with controlled invariants.
• The method employs numerical semigroups, specific scaling factors, and gluing binomials to preserve key properties like non-decreasing Hilbert functions and Cohen–Macaulayness.
• Applications range from categorical constructions and projective closures to computational spline design, enhancing research on moduli spaces and curve degenerations.

Ideal point gluing of curves refers to the process of joining two (or more) algebraic or geometric objects—most commonly curves, their associated semigroups, or their derived categories—at specific points or in an asymptotic regime such that structure, invariants, or categorical data are naturally “glued” through an ideal or prescribed interface. This concept appears in several distinct settings in contemporary algebraic geometry, singularity theory, arithmetic geometry, geometric analysis, and categorical algebra, ranging from explicit semigroup–based constructions of monomial curves to categorical gluings in the derived category framework. Recent research has developed rigorous techniques to classify, construct, and analyze families obtained via ideal point gluing, revealing deep connections with invariants such as Hilbert functions, Cohen–Macaulay and Gorenstein properties, arithmetic torsion, moduli spaces, and compact type degenerations.

1. Algebraic Constructions: Gluing of Numerical Semigroups and Monomial Curves

In the context of algebraic geometry, ideal point gluing is closely linked to the operation of gluing numerical semigroups associated to monomial curves. The method, developed for constructing new families of monomial curves, consists of the controlled amalgamation of two numerical semigroups, $S_1 = \langle m_1, \ldots, m_i \rangle$ and $S_2 = \langle n_1, \ldots, n_k \rangle$, by appropriately choosing scaling factors $p$ and $q$ such that $\gcd(p, q) = 1$ and certain coprimality and minimality conditions are satisfied. The resulting glued semigroup is

$$S = \langle q m_1, \ldots, q m_i, p n_1, \ldots, p n_k \rangle,$$

which encodes the arithmetic of the ideal point gluing. The associated monomial curve corresponds to the parametrization

$$C = \left\{ (t^{q m_1}, \ldots, t^{q m_i}, t^{p n_1}, \ldots, t^{p n_k}) : t \in K \right\}.$$

The defining ideal of such a curve is generated by the union of the generators of $I(C_1)$ and $I(C_2)$ and a "gluing" binomial $x_1^{b_1}\cdots x_i^{b_i} - y_1^{a_1}$, which algebraically links the two curves (Arslan et al., 2011). Special cases such as “nice gluing”—which requires $a_1 \le b_1 + \cdots + b_i$—ensure further control over the singularity and tangent cone structure.

This gluing preserves various properties including the non-decreasing Hilbert function and, under appropriate conditions, Cohen–Macaulayness and Gorensteinness of the tangent cones. Such constructions underpin systematic approaches for producing large families of 1-dimensional Gorenstein local rings supporting Rossi’s conjecture (every Gorenstein local ring has a non-decreasing Hilbert function) (Arslan et al., 2011, Jafari et al., 2013).

2. Hilbert Function, Tangent Cone, and Invariant Transfer Under Gluing

Properties of the glued curves and their associated local rings are intimately controlled by the gluing operation. For instance, non-decreasing Hilbert functions are preserved even when the tangent cone is not Cohen–Macaulay, providing important evidence for Rossi's conjecture (Arslan et al., 2011). When both constituent curves have Cohen–Macaulay tangent cones, the tangent cone of the glued curve also retains this property under nice or specific gluing conditions (Jafari et al., 2013). This transfer of invariants is governed by explicit arithmetic in the semigroup and by the standard bases in the defining ideals.

A crucial criterion for tangent cone Cohen–Macaulayness involves the arithmetic property:

$$\operatorname{ord}_S(s + e) = \operatorname{ord}_S(s) + 1 \quad \forall\, s \in S,$$

which ensures the leading form is a non–zero divisor. For Gorensteinness, the Apéry set symmetry and $M$-purity conditions (all maximal Apéry elements with respect to multiplicity have equal order) are required. In the gluing setting, these invariants can be designed to transfer along one factor, allowing explicit construction of new families with prescribed homological and arithmetic properties (Jafari et al., 2013).

3. Projective Closures, Star Gluing, and Betti Sequence Preservation

Extensions of ideal point gluing in projective geometry include “star gluing” of numerical semigroups, which controls the projective closure behavior of monomial curves. Here, the largest generator corresponds to an “ideal point” in the projective closure, and under specific constraints on the gluing parameters, homological invariants (such as arithmetically Cohen–Macaulay and Gorenstein properties) are preserved throughout the gluing process (Saha et al., 2021).

The Betti sequence of the affine curve and its projective closure are shown to coincide given a Gröbner basis satisfying certain divisibility conditions by the "largest" variable, precisely controlling the influence of the ideal point at infinity. Simple gluing operations admit an additive formula for the change in Betti numbers, enabling systematic generation of curves and semigroups with arbitrarily prescribed Cohen–Macaulay type and minimal free resolutions. This framework provides a robust algebraic setting for engineering glued curves with controlled projective singularity data.

Gluing Type	Preserved Properties	Key Condition
Star Gluing	ACM, Gorenstein, Betti sequence	$\gcd(p, q) = 1$, largest generator constraint
Simple Gluing	Betti sequence change predictable	Additive formula for $\beta_i$
Gröbner Basis	Betti invariants	Largest variable divides support of nonlead terms

4. Arithmetic Gluing: Torsion, Jacobians, and Moduli of Higher-Genus Curves

Ideal point gluing features prominently in arithmetic geometry, particularly in constructing genus-3 curves whose Jacobian is isogenous to the product of the Jacobians of genus-1 and genus-2 curves, glued along their $(\ell, \ell)$-torsion (Hanselman et al., 2020, Saengrungkongka et al., 13 Feb 2025). For primes $\ell \ge 3$, the process involves the identification of indecomposable maximal isotropic subgroups

$$G \subset \operatorname{Jac}(X)[\ell] \times \operatorname{Jac}(Y)[\ell]$$

with constraints ensuring the induced polarization is principal. The gluing yields

$$\operatorname{Jac}(Z) \cong (\operatorname{Jac}(X) \times \operatorname{Jac}(Y))/G$$

up to quadratic twist. Algorithmic advances optimize the search for suitable pairs $(X,Y)$ and the computation of isomorphism classes, leading to explicit constructions of genus-3 curves with rich Sato–Tate groups, rational torsion, and large endomorphism rings (Saengrungkongka et al., 13 Feb 2025).

When gluing along $2$-torsion, both interpolation (numerical analytic) and Kummer variety methods are employed to produce a plane quartic model for $Z$. The latter involves hyperplane sections of the Kummer variety whose normalization relates to $X$, and the quotient construction realizes the isogeny (Hanselman et al., 2020). The arithmetic consequences include the construction of Jacobs over $\mathbb{Q}$ with large rational torsion, e.g., order $70$.

5. Categorical Ideal Point Gluing, Augmentations, and Degenerations

A major categorical approach to ideal point gluing has been formulated via the derived categories of smooth projective curves. The construction is defined by gluing the bounded derived categories $\mathcal{D}^b(C_1)$ and $\mathcal{D}^b(C_2)$ with gluing bimodule

$$\operatorname{RHom}(-,-)_{\text{glue}} = H^*(C_1 \times C_2, (F_1^\vee \boxtimes F_2) \otimes \mathcal{I}_{(x_1,x_2)})$$

where $\mathcal{I}_{(x_1,x_2)}$ is the ideal sheaf of a point in $C_1 \times C_2$ (Alexeev et al., 15 Sep 2025). The resulting triangulated category exhibits new phenomena, notably the existence of “exotic” exceptional objects:

$$E = \operatorname{Cone}(\mathcal{O}_{x_1} \to^\epsilon \mathcal{O}_{x_2})[1],$$

which are not geometrically supported on either curve individually.

The orthogonal complement of $E$ in the glued category yields the “reduced ideal point gluing” category $\operatorname{RPG}(C_1, C_2)$, which is smooth and proper and shares many numerical invariants with the derived category of a smooth curve of genus $g_1 + g_2$, yet is not equivalent to such a category when both $g_1$ and $g_2 \geq 1$. Compact type degeneration phenomena relate the construction to families of curves with nodal central fiber, enriching the connection between categorical geometry and moduli compactifications. This provides a categorification of the boundary stratification in the Deligne–Mumford moduli space of curves.

6. Geometric and Analytic Gluing: CAT(0) Domains and Moduli Spaces

Geometric analysis of ideal point gluing is exemplified by CAT(0) domain gluing. Two locally CAT(0) domains in the Euclidean plane (or subsets of curved surfaces of $k \leq 0$) can be glued along boundary arcs provided the sum of the signed curvatures at each gluing point is nonpositive:

$$\kappa_A(s) + \kappa_B(s) \leq 0,\ \forall s\in\,\text{arc}.$$

This curvature balancing ensures preservation of local CAT(0) geometry, uniqueness of geodesics, and suitable angle conditions, with direct analogues in ideal endpoint gluing regimes (e.g., in Teichmüller theory or moduli of surfaces) (Charitos et al., 4 Apr 2025).

In orbifold stratified spaces, ideal point gluing underpins the construction of smooth structures on moduli spaces of curves, such as the Deligne–Mumford compactification. Via gluing bundles equipped with linear stratification and good gluing atlases obeying sewing and inward-extendibility properties, one achieves compatible orbifold structures vital for understanding moduli spaces of pseudo-holomorphic curves and related degenerations (Chen et al., 2015).

7. Interpolating Space Curves and Computational Aspects

In the context of computational geometry and spline design, ideal point gluing arises in the divide-and-conquer construction of $G^r$—typically $G^2$—smooth interpolating space curves. The methodology relies on local function construction, redistribution (parameter normalization), blending functions with matched vanishing derivatives, and gluing functions to smoothly aggregate curve segments. This ensures geometric continuity, absence of cusps and self-intersections, preservation of convexity and boundary properties, and sphere preservation in specific inputs (Hu et al., 17 May 2024). The algorithmic steps are implemented in practical software (MATLAB), and numerical examples confirm the efficacy and versatility of the approach.

Conclusion

Ideal point gluing of curves spans a vast constellation of modern mathematical techniques, unifying explicit algebraic constructions, arithmetic algorithms, geometric analysis, categorical resolutions of singularities, and computational methodologies. The core principle remains the deliberate, invariant–preserving amalgamation of two or more objects at prescribed points, interfaces, or asymptotic regimes, as encoded through semigroup generators, torsion modules, sheaf data, or curvature. These constructions are instrumental in advancing research on singularity theory, moduli spaces, arithmetic and geometric properties of curves, categorical frameworks, and practical design algorithms, providing fertile ground for further generalizations and applications.