  Let $\Lambda$ be a finite dimensional algebra over an algebraically closed
field, and ${\Bbb S}$ a finite sequence of simple left $\Lambda$-modules. In
[6, 9], quasiprojective algebraic varieties with accessible affine open covers
were introduced, for use in classifying the uniserial representations of
$\Lambda$ having sequence ${\Bbb S}$ of consecutive composition factors. Our
principal objectives here are threefold: One is to prove these varieties to be
`good approximations' -- in a sense to be made precise -- to geometric
quotients of the classical varieties $\operatorname{Mod-Uni}({\Bbb S})$
parametrizing the pertinent uniserial representations, modulo the usual
conjugation action of the general linear group. To some extent, this fills the
information gap left open by the frequent non-existence of such quotients. A
second goal is that of facilitating the transfer of information among the
`host' varieties into which the considered uniserial varieties can be embedded.
These tools are then applied towards the third objective, concerning the
existence of geometric quotients: We prove that $\operatorname{Mod-Uni}({\Bbb
S})$ has a geometric quotient by the $GL$-action precisely when the uniserial
variety has a geometric quotient modulo a certain natural algebraic group
action, in which case the two quotients coincide. Our main results are
exploited in a representation-theoretic context: Among other consequences, they
yield a geometric characterization of the algebras of finite uniserial type
which supplements existing descriptions, but is cleaner and more readily
checkable.
