The geometry of uniserial representations of algebras II. Alternate viewpoints and uniqueness

We provide two alternate settings for a family of varieties modeling the uniserial representations with fixed sequence of composition factors over a finite dimensional algebra. The first is a quasi-projective subvariety of a Grassmannian containing the members of the mentioned family as a principal affine open cover; among other benefits, one derives invariance from this intrinsic description. The second viewpoint re-interprets the `uniserial varieties' as locally closed subvarieties of the traditional module varieties; in particular, it exhibits closedness of the fibres of the canonical maps from the uniserial varieties to the uniserial representations.