  We give a local characterization for when certain quiver representations in
semisimple Abelian categories are semisimple, among them those arising from
degenerations of linear series. This paper is the first of two, aimed to
describe all the schematic limits of families of divisors associated to a given
family of linear series on a one-dimensional family of projective varieties
degenerating to a connected reduced projective scheme $X$ defined over any
field, under the assumption that the total space of the family is regular along
$X$, by means of certain quiver Grassmannians.
