Hurwitz moduli varieties parameterizing pointed covers of an algebraic curve with a fixed monodromy group
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911221499822080 |
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| author | Kanev, Vassil |
| author_facet | Kanev, Vassil |
| contents | Given a smooth, projective curve $Y$, a point $y_0 \in Y$, a positive integer $n$, and a transitive subgroup $G$ of the symmetric group $S_{d}$ we study smooth, proper families, parameterized by algebraic varieties, of pointed degree $d$ covers of $(Y,y_0)$, $(X,x_{0})\to (Y,y_0)$, branched in $n$ points of $Y\setminus y_{0}$, whose monodromy group equals $G$. We construct a Hurwitz space $H$, an algebraic variety whose points are in bijective correspondence with the equivalence classes of pointed covers of $(Y,y_0)$ of this type. We construct explicitly a family parameterized by $H$, whose fibers belong to the corresponding equivalence classes, and prove that it is universal. We use classical tools of algebraic topology and of complex algebraic geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12756 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hurwitz moduli varieties parameterizing pointed covers of an algebraic curve with a fixed monodromy group Kanev, Vassil Algebraic Geometry 14H30 (Primary) 14H10, 14D22 (Secondary) Given a smooth, projective curve $Y$, a point $y_0 \in Y$, a positive integer $n$, and a transitive subgroup $G$ of the symmetric group $S_{d}$ we study smooth, proper families, parameterized by algebraic varieties, of pointed degree $d$ covers of $(Y,y_0)$, $(X,x_{0})\to (Y,y_0)$, branched in $n$ points of $Y\setminus y_{0}$, whose monodromy group equals $G$. We construct a Hurwitz space $H$, an algebraic variety whose points are in bijective correspondence with the equivalence classes of pointed covers of $(Y,y_0)$ of this type. We construct explicitly a family parameterized by $H$, whose fibers belong to the corresponding equivalence classes, and prove that it is universal. We use classical tools of algebraic topology and of complex algebraic geometry. |
| title | Hurwitz moduli varieties parameterizing pointed covers of an algebraic curve with a fixed monodromy group |
| topic | Algebraic Geometry 14H30 (Primary) 14H10, 14D22 (Secondary) |
| url | https://arxiv.org/abs/2403.12756 |