The Picard Group of the Stack of Pointed Smooth Cyclic Covers of the Projective Line
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916720728342528 |
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| author | Landi, Alberto |
| author_facet | Landi, Alberto |
| contents | We study the stack $\mathcal{H}_{r,g,n}$ of $n$-pointed smooth cyclic covers of degree $r$ between smooth curves of genus $g$ and the projective line. We give two presentations of an open substack of $\mathcal{H}_{r,g,n}$ as a quotient stack, and we study its complement. Using this, we compute the integral Picard group of $\mathcal{H}_{r,g,n}$. Moreover, we obtain a very explicit description of the generators of the Picard group, which have evident geometric meaning. As a corollary of the computation, we get the integral Picard group of the stack $\mathcal{H}_{g,n}$ of $n$-pointed hyperelliptic curves of genus $g$. Finally, taking $g=2$ and recalling that $\mathcal{H}_{2,n}=\mathcal{M}_{2,n}$, we obtain $\mathrm{Pic}(\mathcal{M}_{2,n})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20045 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Picard Group of the Stack of Pointed Smooth Cyclic Covers of the Projective Line Landi, Alberto Algebraic Geometry 14D23, 14C22, 14H10 We study the stack $\mathcal{H}_{r,g,n}$ of $n$-pointed smooth cyclic covers of degree $r$ between smooth curves of genus $g$ and the projective line. We give two presentations of an open substack of $\mathcal{H}_{r,g,n}$ as a quotient stack, and we study its complement. Using this, we compute the integral Picard group of $\mathcal{H}_{r,g,n}$. Moreover, we obtain a very explicit description of the generators of the Picard group, which have evident geometric meaning. As a corollary of the computation, we get the integral Picard group of the stack $\mathcal{H}_{g,n}$ of $n$-pointed hyperelliptic curves of genus $g$. Finally, taking $g=2$ and recalling that $\mathcal{H}_{2,n}=\mathcal{M}_{2,n}$, we obtain $\mathrm{Pic}(\mathcal{M}_{2,n})$. |
| title | The Picard Group of the Stack of Pointed Smooth Cyclic Covers of the Projective Line |
| topic | Algebraic Geometry 14D23, 14C22, 14H10 |
| url | https://arxiv.org/abs/2310.20045 |