On the Prym map of degree 4 cyclic covers of hyperelliptic curves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912981027127296 |
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| author | Shatsila, Anatoli |
| author_facet | Shatsila, Anatoli |
| contents | In this paper, we study the Prym map associated to degree 4 étale cyclic covers of genus $g$ hyperelliptic curves restricted to the irreducible component $\mathcal{RH}_g[4]^{hyp}$ of the moduli space of such covers where an intermediate cover is hyperelliptic. We show that for $g \geq 3$ the Prym map is injective on $\mathcal{RH}_g[4]^{hyp}$. In the case $g=2$ (where $\mathcal{RH}_2[4]^{hyp} = \mathcal{RH}_2[4]$) we prove that non-empty fibers of the Prym map, apart from two exceptional fibers, are isomorphic to the projective line without 8 points. Moreover, we obtain a new description of the space $\mathcal{RH}_g[4]^{hyp}$ in terms of tuples of complex numbers and find equations of hyperelliptic curves arising from such covers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_20838 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Prym map of degree 4 cyclic covers of hyperelliptic curves Shatsila, Anatoli Algebraic Geometry 14H30, 14H40, 14H45, 14H55, 14K12 In this paper, we study the Prym map associated to degree 4 étale cyclic covers of genus $g$ hyperelliptic curves restricted to the irreducible component $\mathcal{RH}_g[4]^{hyp}$ of the moduli space of such covers where an intermediate cover is hyperelliptic. We show that for $g \geq 3$ the Prym map is injective on $\mathcal{RH}_g[4]^{hyp}$. In the case $g=2$ (where $\mathcal{RH}_2[4]^{hyp} = \mathcal{RH}_2[4]$) we prove that non-empty fibers of the Prym map, apart from two exceptional fibers, are isomorphic to the projective line without 8 points. Moreover, we obtain a new description of the space $\mathcal{RH}_g[4]^{hyp}$ in terms of tuples of complex numbers and find equations of hyperelliptic curves arising from such covers. |
| title | On the Prym map of degree 4 cyclic covers of hyperelliptic curves |
| topic | Algebraic Geometry 14H30, 14H40, 14H45, 14H55, 14K12 |
| url | https://arxiv.org/abs/2508.20838 |