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Let $\Lambda$ be a finite dimensional algebra over an algebraically closedfield, and ${\Bbb S}$ a finite sequence of simple left $\Lambda$-modules. In[6, 9], quasiprojective algebraic varieties with accessible affine open coverswere introduced, for use in classifying the uniserial representations of$\Lambda$ having sequence ${\Bbb S}$ of consecutive composition factors. Ourprincipal objectives here are threefold: One is to prove these varieties to be`good approximations' -- in a sense to be made precise -- to geometricquotients of the classical varieties $\operatorname{Mod-Uni}({\Bbb S})$parametrizing the pertinent uniserial representations, modulo the usualconjugation action of the general linear group. To some extent, this fills theinformation gap left open by the frequent non-existence of such quotients. Asecond 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 theexistence of geometric quotients: We prove that $\operatorname{Mod-Uni}({\BbbS})$ has a geometric quotient by the $GL$-action precisely when the uniserialvariety has a geometric quotient modulo a certain natural algebraic groupaction, in which case the two quotients coincide. Our main results areexploited in a representation-theoretic context: Among other consequences, theyyield a geometric characterization of the algebras of finite uniserial typewhich supplements existing descriptions, but is cleaner and more readilycheckable.

Varieties of uniserial representations IV. Kinship to geometric quotients