$\mathbb{A}^1$-connected stacky curves and the Brauer group of moduli of elliptic curves
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913599938625536 |
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| author | Deshmukh, Neeraj Yadav, Suraj |
| author_facet | Deshmukh, Neeraj Yadav, Suraj |
| contents | Given a smooth scheme X with an action by an affine algebraic group G, we give a formula to compute the Nisnevich sheaf of the motivic connected components of the quotient stack [X/G] in the case of an orbifold. We apply it to identify all the $\mathbb{A}^1$-connected stacky curves and prove the $\mathbb{A}^1$ -connectedness of the moduli stack of elliptic curves. We also prove homotopy purity for the smooth algebraic stacks and recover a computation of the Picard group of stacky orbifold curves by computing their mixed motive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01525 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathbb{A}^1$-connected stacky curves and the Brauer group of moduli of elliptic curves Deshmukh, Neeraj Yadav, Suraj Algebraic Geometry Given a smooth scheme X with an action by an affine algebraic group G, we give a formula to compute the Nisnevich sheaf of the motivic connected components of the quotient stack [X/G] in the case of an orbifold. We apply it to identify all the $\mathbb{A}^1$-connected stacky curves and prove the $\mathbb{A}^1$ -connectedness of the moduli stack of elliptic curves. We also prove homotopy purity for the smooth algebraic stacks and recover a computation of the Picard group of stacky orbifold curves by computing their mixed motive. |
| title | $\mathbb{A}^1$-connected stacky curves and the Brauer group of moduli of elliptic curves |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2410.01525 |