We describe all schematic limits of families of divisors associated to a given family of rank-r 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. More precisely, the degenerating family gives rise to a special quiver Q, called a \(\mathbb {Z}^n\) -quiver, where \(n+1\) is the number of irreducible components of X, a special representation \({\mathfrak {L}}\) of Q in the category of line bundles over X, called a maximal exact linked net, and a special subrepresentation \({\mathfrak {V}}\) of the representation \(H^0(X,{\mathfrak {L}})\) induced from \({\mathfrak {L}}\) by taking global sections, called a pure exact finitely generated linked net of dimension \(r+1\) . Given \({\mathfrak {g}}=(Q,{\mathfrak {L}},{\mathfrak {V}})\) satisfying these properties, we prove that the quiver Grassmanian \(\mathbb{L}\mathbb{P}(\mathfrak {V})\) of subrepresentations of \(\mathfrak {V}\) of pure dimension 1, called a linked projective space, is local complete intersection, reduced and of pure dimension r. Furthermore, we prove that there is a morphism \(\mathbb{L}\mathbb{P}(\mathfrak {V})\rightarrow \text {Hilb}_X\) , and that its image parameterizes all schematic limits of divisors along the degenerating family of linear series if \({\mathfrak {g}}\) arises from one.
