  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.
