Tracking the variety of interleavings

In topological data analysis persistence modules are used to distinguish the legitimate topological features of a finite data set from noise. Interleavings between persistence modules feature prominantly in the analysis. One can show that for $\epsilon$ positive, the collection of $\epsilon$-interleavings between two persistence modules $M$ and $N$ has the structure of an affine variety, Thus, the smallest value of $\epsilon$ corresponding to a nonempty variety is the interleaving distance. With this in mind, it is natural to wonder how this variety changes with the value of $\epsilon$, and what information about $M$ and $N$ can be seen from just the knowledge of their varieties. In this paper, we focus on the special case where $M$ and $N$ are interval modules. In this situation we classify all possible progressions of varieties, and determine what information about $M$ and $N$ is present in the progression.