  We describe all the 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 \emph{$\mathbb{Z}^n$-quiver}, a special
representation $\mathfrak L$ of $Q$ in the category of line bundles over $X$,
called a \emph{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 \emph{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 $\mathbb{LP}(\mathfrak{V})$ of subrepresentations of $\mathfrak{V}$
of pure dimension 1, called a \emph{linked projective space}, is local complete
intersection, reduced and of pure dimension $r$. Furthermore, we prove that
there is a morphism $\mathbb{LP}(\mathfrak{V})\to\text{Hilb}_X$, and that its
image parameterizes all the schematic limits of divisors along the degenerating
family of linear series if $\mathfrak g$ arises from one.
