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.
Categories: math.AG