Stabilization of codimension of persistence barcodes

Abstract

Given a pointwise finite-dimensional persistence module over a totally ordered set $S$, a theorem of Crawley-Boevey guarantees the existence of a barcode. When the set $S$ is finite, the persistence module is an equioriented type-A quiver representation and the barcode identifies a distinguished point in an algebraic variety. We prove a formula for the codimension of this variety inside an ambient space of comparable representations which depends only on the combinatorics of the barcode. We therefore extend the notion of codimension to persistence modules over any set $S$ and prove that this extension is well-defined and can be effectively computed via a stabilization property of approximating quiver representations. Further, we prove realization theorems by constructing explicit examples via persistent homology of data for which the codimension can realize any natural number as its value.