  It is shown that, given any finite dimensional, split basic algebra $\Lambda
= K\Gamma/I$ (where $\Gamma$ is a quiver and $I$ an admissible ideal in the
path algebra $K \Gamma$), there is a finite list of affine algebraic varieties,
the points of which correspond in a natural fashion to the isomorphism types of
uniserial left $\Lambda$-modules, and the geometry of which faithfully reflects
the constraints met in constructing such modules. A constructive coordinatized
access to these varieties is given, as well as to the accompanying natural
surjections from the varieties onto families of uniserial modules with fixed
composition series. The fibres of these maps are explored, one of the results
being a simple algorithm to resolve the isomorphism problem for uniserial
modules. Moreover, new invariants measuring the complexity of the uniserial
representation theory are derived from the geometric viewpoint. Finally, it is
proved that each affine algebraic variety arises as a variety of uniserial
modules over a suitable finite dimensional algebra, in a setting where the
points are in one-one correspondence with the isomorphism classes of uniserial
modules.
