The Integral Chow Rings of the Moduli Stacks of Hyperelliptic Prym Pairs III
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arXiv
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| Natura: | Preprint |
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2025
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| author | Cela, Alessio Landi, Alberto |
| author_facet | Cela, Alessio Landi, Alberto |
| contents | This paper is the third and final part of a series devoted to the description of the integral Chow rings of the moduli stacks of hyperelliptic Prym pairs. For a fixed genus $g$, there are two natural stacks, $\mathcal{RH}_g$ and $\widetilde{\mathcal{RH}}_g$, parametrizing hyperelliptic Prym pairs, with the former being the $μ_2$-rigidification of the latter. Both decompose as the disjoint union of $\lfloor (g+1)/2 \rfloor$ components, denoted $\mathcal{RH}_g^n$ and $\widetilde{\mathcal{RH}}_g^n$ for $n = 1, \ldots, \lfloor (g+1)/2 \rfloor$. In this paper we present quotient stack descriptions of the components $\mathcal{RH}_g^n$ for even $g$ and compute their integral Chow rings, thereby completing the computation for all irreducible components of $\mathcal{RH}_g$. In addition, we give quotient stack presentations for all irreducible components of $\widetilde{\mathcal{RH}}_g$ and determine when the rigidification map $\widetilde{\mathcal{RH}}_g^n \to \mathcal{RH}_g^n$ is a root gerbe. We then use this to compute the Chow rings of $\widetilde{\mathcal{RH}}_g^n$ for all $g$ and $n$, with the sole exception of the case where $g$ is odd and $n=(g+1)/2$.
Finally, in the appendix, we discuss $G$-gerbes induced by an homomorphism of abelian groups $H \to G$ and an $H$-gerbe. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15186 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Integral Chow Rings of the Moduli Stacks of Hyperelliptic Prym Pairs III Cela, Alessio Landi, Alberto Algebraic Geometry 14C15 (Primary), 14D23, 14H40 (Secondary) This paper is the third and final part of a series devoted to the description of the integral Chow rings of the moduli stacks of hyperelliptic Prym pairs. For a fixed genus $g$, there are two natural stacks, $\mathcal{RH}_g$ and $\widetilde{\mathcal{RH}}_g$, parametrizing hyperelliptic Prym pairs, with the former being the $μ_2$-rigidification of the latter. Both decompose as the disjoint union of $\lfloor (g+1)/2 \rfloor$ components, denoted $\mathcal{RH}_g^n$ and $\widetilde{\mathcal{RH}}_g^n$ for $n = 1, \ldots, \lfloor (g+1)/2 \rfloor$. In this paper we present quotient stack descriptions of the components $\mathcal{RH}_g^n$ for even $g$ and compute their integral Chow rings, thereby completing the computation for all irreducible components of $\mathcal{RH}_g$. In addition, we give quotient stack presentations for all irreducible components of $\widetilde{\mathcal{RH}}_g$ and determine when the rigidification map $\widetilde{\mathcal{RH}}_g^n \to \mathcal{RH}_g^n$ is a root gerbe. We then use this to compute the Chow rings of $\widetilde{\mathcal{RH}}_g^n$ for all $g$ and $n$, with the sole exception of the case where $g$ is odd and $n=(g+1)/2$. Finally, in the appendix, we discuss $G$-gerbes induced by an homomorphism of abelian groups $H \to G$ and an $H$-gerbe. |
| title | The Integral Chow Rings of the Moduli Stacks of Hyperelliptic Prym Pairs III |
| topic | Algebraic Geometry 14C15 (Primary), 14D23, 14H40 (Secondary) |
| url | https://arxiv.org/abs/2509.15186 |