Abelian covers and second fundamental form
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866910380240928768 |
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| author | Frediani, Paola |
| author_facet | Frediani, Paola |
| contents | We give some conditions on a family of abelian covers of ${\mathbb P}^1$ of genus $g$ curves, that ensure that the family yields a subvariety of ${\mathsf A}_g$ which is not totally geodesic, hence it is not Shimura. As a consequence, we show that for any abelian group $G$, there exists an integer $M$ which only depends on $G$ such that if $g >M$, then the family yields a subvariety of ${\mathsf A}_g$ which is not totally geodesic. We prove then analogous results for families of abelian covers of ${\tilde C}_t \rightarrow {\mathbb P}^1 = {\tilde C}_t/{\tilde G}$ with an abelian Galois group ${\tilde G}$ of even order, proving that under some conditions, if $σ\in {\tilde G}$ is an involution, the family of Pryms associated with the covers ${\tilde C}_t \rightarrow C_t= {\tilde C}_t/\langle σ\rangle$ yields a subvariety of ${\mathsf A}_{p}^δ$ which is not totally geodesic. As a consequence, we show that if ${\tilde G} =({\mathbb Z}/N{\mathbb Z})^m$ with $N$ even, and $σ$ is an involution in ${\tilde G}$, there exists an integer $M(N)$ which only depends on $N$ such that, if ${\tilde g} = g({\tilde C}_t) > M(N)$, then the subvariety of the Prym locus in ${\mathsf A}^δ_{p}$ induced by any such family is not totally geodesic (hence it is not Shimura). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_07947 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Abelian covers and second fundamental form Frediani, Paola Algebraic Geometry 14H40, 14H10 We give some conditions on a family of abelian covers of ${\mathbb P}^1$ of genus $g$ curves, that ensure that the family yields a subvariety of ${\mathsf A}_g$ which is not totally geodesic, hence it is not Shimura. As a consequence, we show that for any abelian group $G$, there exists an integer $M$ which only depends on $G$ such that if $g >M$, then the family yields a subvariety of ${\mathsf A}_g$ which is not totally geodesic. We prove then analogous results for families of abelian covers of ${\tilde C}_t \rightarrow {\mathbb P}^1 = {\tilde C}_t/{\tilde G}$ with an abelian Galois group ${\tilde G}$ of even order, proving that under some conditions, if $σ\in {\tilde G}$ is an involution, the family of Pryms associated with the covers ${\tilde C}_t \rightarrow C_t= {\tilde C}_t/\langle σ\rangle$ yields a subvariety of ${\mathsf A}_{p}^δ$ which is not totally geodesic. As a consequence, we show that if ${\tilde G} =({\mathbb Z}/N{\mathbb Z})^m$ with $N$ even, and $σ$ is an involution in ${\tilde G}$, there exists an integer $M(N)$ which only depends on $N$ such that, if ${\tilde g} = g({\tilde C}_t) > M(N)$, then the subvariety of the Prym locus in ${\mathsf A}^δ_{p}$ induced by any such family is not totally geodesic (hence it is not Shimura). |
| title | Abelian covers and second fundamental form |
| topic | Algebraic Geometry 14H40, 14H10 |
| url | https://arxiv.org/abs/2105.07947 |