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| Main Authors: | , , |
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| Format: | Preprint |
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2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2112.00514 |
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| _version_ | 1866914222195081216 |
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| author | Esteves, Eduardo Santos, Renan Vital, Eduardo |
| author_facet | Esteves, Eduardo Santos, Renan Vital, Eduardo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_00514 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quiver representations arising from degenerations of linear series, II Esteves, Eduardo Santos, Renan Vital, Eduardo Algebraic Geometry 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. |
| title | Quiver representations arising from degenerations of linear series, II |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2112.00514 |