The irreducible components of the varieties parametrizing the representations with fixed dimension of a finite dimensional algebra $A$ (equivalently, the representations of the Gabriel quiver and relations of $A$) are explored, in terms of both their geometry and the structure of the modules they encode. In particular, given an irreducible component $C$ of such a variety, we establish existence and uniqueness (in a sense to be specified) of modules which display all generic ``categorical" properties of the representations corresponding to the points of $C$; here ``categorical" means ``stable under self-equivalences of the category of $A$-modules". We then use this result towards an investigation of the generic modules for path algebras modulo relations. This is done via a back and forth between the classical affine parametrizing varieties and projective alternates inside suitable Grassmann varieties.