Degenerations of families of bands and strings for gentle algebras

TL;DR

This work analyzes degenerations of families of string and band modules over gentle algebras by introducing a combinatorial degeneration framework tied to $h$-vectors. It shows that each family $ ext{O}_{oldsymbol{D}}$ is characterized by a fixed $h$-vector and that degenerations imply a partial $h$-order, with two concrete degeneration mechanisms: deleting an arrow and resolving reachings. The authors establish that resolving reachings yields degenerations and provide irreducibility results for the families, complemented by explicit examples (notably for the Kronecker quiver). The results offer a concrete, combinatorial pathway to study the component structure of module varieties for gentle algebras and suggest natural extensions to broader string algebras and degeneration theory in representation varieties.

Abstract

Let $A$ be a gentle algebra. For every collection of string and band diagrammes, we consider the constructible subset of the variety of representations containing all modules with this underlying diagramme. We study degenerations of such sets. We show that these sets are defined by vectors of integers which we call $h$-vectors and which are related to a restricted version of the hom-order. We provide combinatorial criteria for the existence of a degeneration, involving the removal of an arrow or the resolving of a type of configuration called "reaching".

Figures (17)

• Figure 1: Notation of substrings.
• Figure 2:
• Figure 3: Module orbits for the dimension vector $(1,1)$.
• Figure 4: Family of the band $ba^-$.
• Figure 5: The partial order of degenerations of strings and minimal bands in dimension $(1,1)$. The vectors represent the $h$-vector.

Theorems & Definitions (119)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Example 2.4
• Remark 2.5
• Definition 2.6
• Example 2.8
• Definition 2.9
• Example 2.10
• Definition 2.11