Abstract: We describe all the schematic limits of divisors associated to any family of linear series on any one- dimensional family of projective varieties degenerating to any connected reduced projective scheme X defined over any field, under the assumption that the total space of the family is regular along X. More precisely, the degenerating family gives rise to a special quiver Q, called a Zn-quiver, a special representation L of Q in the category of line bundles over X, called a maximal exact linked net, and a special subrepresentation V of the representation induced from L by taking global sections, called a pure exact finitely generated linked net. Given g = (Q, L, V ) satisfying these properties, we prove that the quiver Grassmanian G of subrepresentations of V of pure dimension 1, called a linked projective space, is Cohen-Macaulay, reduced and of pure dimension. Furthermore, we prove that there is a morphism from G to the Hilbert scheme of X whose image parameterizes all the schematic limits of divisors along the degenerating family of linear series if g arises from one. Joint work with Eduardo Vital and Renan Santos.