Quiver representations arising from degenerations of linear series, II

Abstract

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.

arXiv comment36 pages

References in corpus (1)

• Quiver representations arising from degenerations of linear series, I

Cited by in corpus (1)

• Quiver representations arising from degenerations of linear series, I