Degenerations of families of bands and strings for gentle algebras

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”.