Quiver representations arising from degenerations of linear series, I

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.