Degenerations of families of bands and strings for gentle algebras

TL;DR

This paper investigates the degenerations of families of modules over gentle algebras, introducing combinatorial criteria and $h$-vectors to understand when one family degenerates into another.

Contribution

It introduces $h$-vectors and combinatorial criteria for degenerations of modules over gentle algebras, linking geometric and combinatorial aspects.

Findings

$h$-vectors characterize degenerations.

Degeneration criteria involve arrow removal and configuration resolution.

Provides a combinatorial framework for understanding module degenerations.

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".