Reduced Ideal Point Gluing in Derived Categories

• Reduced ideal point gluing is a categorical operation that joins the derived categories of two smooth projective curves along specified points using an ideal sheaf.
• The process introduces an exotic exceptional object from mixed skyscraper sheaves, whose removal via right orthogonality refines the triangulated structure.
• This construction plays a key role in modeling compact type degenerations, equipping researchers with computable invariants and semiorthogonal decompositions.

Reduced ideal point gluing of curves is a categorical construction for assembling the derived categories of two smooth projective curves along specified points, followed by a reduction whereby an exotic exceptional object resulting from the gluing is removed. This operation produces triangulated categories encapsulating both the geometric data of the underlying curves and the subtle homological features associated with their interaction at points—especially in families arising in compact type degenerations, where nodal reducible curves appear as central fibers and smooth curves as general fibers. The notion and its detailed properties, including connections to semiorthogonal decomposition, compact type degeneration, and categorical invariants, have been studied in recent literature (Alexeev et al., 15 Sep 2025).

1. Categorical Gluing via Ideals

Given two smooth projective curves $C_1$ and $C_2$ over a field, along with points $x_1 \in C_1$ and $x_2 \in C_2$, the ideal point gluing is implemented on the level of derived categories. The construction proceeds as follows:

• The derived categories $D^b(C_1)$ and $D^b(C_2)$ are glued along the points via a bimodule structure given by the ideal sheaf $\mathcal{I}_{(x_1, x_2)}$ of the point $(x_1, x_2) \in C_1 \times C_2$.
• For objects $\mathcal{F}_1 \in D^b(C_1)$ and $\mathcal{F}_2 \in D^b(C_2)$, morphisms in the glued category are defined by:

$$\operatorname{RHom}_{D}( \mathcal{F}_1, \mathcal{F}_2 ) = H^\bullet( C_1 \times C_2, (\mathcal{F}_1^\vee \boxtimes \mathcal{F}_2) \otimes \mathcal{I}_{(x_1, x_2)} )$$

• The resulting category $D^b(C_1, C_2)$ incorporates the geometric data of both curves as well as the interaction encoded in the chosen pair of points.

This process generalizes standard categorical gluings (by the structure sheaf) and distinguishes itself by the use of the ideal sheaf at the point, introducing new components into the Ext groups.

2. Emergence of Exotic Exceptional Objects

In the context of ideal point gluing via the ideal sheaf, the category $D^b(C_1, C_2)$ typically contains an unexpected ("exotic") exceptional object, denoted $E$. This arises from the mixed behavior of skyscraper sheaves at the glued points:

• $E$ is defined as:

$$E \coloneqq (\mathcal{O}_{x_1}, \mathcal{O}_{x_2}, \varepsilon) = \operatorname{Cone}( \mathcal{O}_{x_1} \xrightarrow{\varepsilon} \mathcal{O}_{x_2} )[1]$$

where $\varepsilon$ is a canonical element in $$H^0( \operatorname{RHom}( \mathcal{O}_{x_1}, \mathcal{O}_{x_2} ) )$$.

• One computes:

$$\operatorname{RHom}(\mathcal{O}_{x_1}, \mathcal{O}_{x_2}) \cong \mathbb{k} \oplus \mathbb{k}[-1] \oplus \mathbb{k}[-1]$$

• The exceptional object $E$ satisfies $\operatorname{Ext}^i(E, E) = 0$ for $i \neq 0$ and $\operatorname{Hom}(E, E) \cong \mathbb{k}$.

This exotic object is inherently tied to the failure of functions to extend across the glued points and is absent for gluings via the structure sheaf.

3. Reduction: Orthogonal Complements and the Reduced Category

The central idea of reduced ideal point gluing is to remove the contribution of $E$ to focus on the nontrivial categorical geometry produced by gluing. The construction is as follows:

• The reduced ideal point gluing ("RPG") category is defined as the right orthogonal to $E$:

$$\operatorname{RPG}(C_1, C_2) := {}^\perp E \subset D^b(C_1, C_2)$$

• The original category admits a semiorthogonal decomposition:

$$D^b(C_1, C_2) = \langle D^b(\mathbb{k}), \operatorname{RPG}(C_1, C_2) \rangle$$

where $D^b(\mathbb{k})$ is the derived category of a point, generated by $E$.

• The role of the reduction is to yield a new invariant triangulated category that is more closely associated with the geometric nature of the glued curves, omitting the trivial effects of $E$.

This treatment enables finer control over the categorical invariants of the construction and clarifies the relationship to augmentation (gluing of a curve to a point via the structure sheaf), which arises as a special case.

4. Homological Invariants and Structure

Basic invariants of $\operatorname{RPG}(C_1, C_2)$ are computed to manifest the nontriviality of the construction:

• Hochschild homology:

$$HH_\bullet(\operatorname{RPG}(C_1, C_2)) = \mathbb{k}^{g_1+g_2}[1] \oplus \mathbb{k}^3 \oplus \mathbb{k}^{g_1+g_2}[-1]$$

where $g_i$ is the genus of $C_i$.

• Numerical Grothendieck group:

$$K^\mathrm{num}(\operatorname{RPG}(C_1, C_2)) \cong \mathbb{Z}^3$$

• If either $C_1$ or $C_2$ has genus zero, RPG is equivalent to the augmentation of the other curve; otherwise, RPG is categorically distinct from simple augmentation.

These invariants distinguish the reduced category from both the full glued category and naive products.

5. Compact Type Degenerations and Gluing in Families

A notable geometric occurrence of reduced ideal point gluing appears in degenerations of curves of compact type:

• Let $\mathcal{C} \to B$ be a proper flat family over a base $B$, with smooth general fiber $\mathcal{C}_b$ ($b \neq o$) and central fiber $\mathcal{C}_o$ a nodal curve $C_1 \cup C_2$.
• Construct a family of derived categories $\mathcal{D} \to B$ such that:
  • $$\mathcal{D}_b \cong D^b(\mathcal{O}, \mathcal{C}_b)$$—augmentation of the smooth fiber.
  • $\mathcal{D}_o \cong \operatorname{RPG}(C_1, C_2)$—reduced ideal point gluing for the nodal fiber.
• The reduced category captures the categorical limit as the smooth curve degenerates to a nodal curve, providing a triangulated category matching the expected geometric behavior.

This framework naturally relates categorical phenomena (exceptional objects, semiorthogonal decompositions) to geometric transitions and is crucial for understanding the categorical invariants of Jacobian degenerations and mapping spaces.

6. Connections and Generalizations

While the reduced ideal point gluing construction arises in the context of derived categories of curves, related principles and analogies are found in several domains:

• In commutative algebra, the process of gluing minimal prime ideals in a local ring (Colbert et al., 2021) implements a (reduced) identification of components of the spectrum, paralleling the categorical identification performed here.
• Numerical semigroup gluing in the construction of monomial curves encapsulates ideal-theoretic control similar to that required for categorical gluing (Arslan et al., 2011, Jafari et al., 2013).
• Deformation-theoretic gluing of Brill–Noether curves (Larson, 2016) and connectedness results for moduli spaces (Bozlee, 10 Jul 2024) underscore the importance of gluing constructions in both the classical and derived categories, notably where points or singularities are "reduced" or identified in moduli settings.

A plausible implication is that reduced ideal point gluing provides a unifying categorical apparatus for studying degeneration, stratification, and moduli of curves, particularly in the context of semiorthogonal decompositions, exceptional collections, and invariants computable via Hochschild or Grothendieck theories.

7. Summary of Key Formulas

$$\begin{aligned} \text{Glued Hom:}\quad &\operatorname{RHom}_{D^b(C_1,C_2)}(\mathcal{F}_1, \mathcal{F}_2) \cong H^\bullet(C_1 \times C_2,\; \mathcal{F}_1^\vee \boxtimes \mathcal{F}_2 \otimes \mathcal{I}_{(x_1,x_2)}) \ \text{Exotic exceptional:}\quad &E \coloneqq (\mathcal{O}_{x_1}, \mathcal{O}_{x_2}, \varepsilon) = \operatorname{Cone}(\mathcal{O}_{x_1} \xrightarrow{\varepsilon} \mathcal{O}_{x_2})[1] \ \text{Reduced category:}\quad &\operatorname{RPG}(C_1, C_2) = {}^\perp E \subset D^b(C_1, C_2) \ \text{Semiorthogonal decomposition:}\quad &D^b(C_1, C_2) = \langle D^b(\mathbb{k}), \operatorname{RPG}(C_1, C_2) \rangle \ \text{Hochschild homology:}\quad &HH_\bullet(\operatorname{RPG}(C_1, C_2)) = \mathbb{k}^{g_1+g_2}[1] \oplus \mathbb{k}^3 \oplus \mathbb{k}^{g_1+g_2}[-1] \end{aligned}$$

The reduced ideal point gluing of curves thus establishes a systematic approach in the theory of derived categories for capturing both global and local features of glued geometric objects, particularly at the intersection of nodal degenerations, categorical invariants, and semiorthogonal structures.