Given a pointwise finite-dimensional persistence module over a totally ordered set
, a
theorem of Crawley-Boevey guarantees the existence of a barcode. When the set
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
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.
Keywords
persistence, barcode, quiver representation, topological
data analysis
