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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
| Publié: |
2023
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2310.15076 |
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| _version_ | 1866910106092830720 |
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| author | Arena, Veronica Di Lorenzo, Andrea Inchiostro, Giovanni Mathur, Siddharth Obinna, Stephen Pernice, Michele |
| author_facet | Arena, Veronica Di Lorenzo, Andrea Inchiostro, Giovanni Mathur, Siddharth Obinna, Stephen Pernice, Michele |
| contents | We establish a criterion for determining when a smooth Deligne-Mumford stack is a weighted blow-up. More precisely, given a smooth Deligne-Mumford stack $\mathcal{X}$ and a Cartier divisor $\mathcal{E} \subset \mathcal{X}$ such that (1) $\mathcal{E}$ is a weighted projective bundle over a smooth Deligne-Mumford stack $\mathcal{Y}$ and (2) for every $y\in\mathcal{Y}$ we have $\mathcal{O}_{\mathcal{X}}(\mathcal{E})|_{\mathcal{E}_y}\simeq \mathcal{O}_{\mathcal{E}_y}(-1)$, then there exists a contraction $\mathcal{X}\to\mathcal{Z}$ to a smooth Deligne-Mumford stack $\mathcal{Z}$. Moreover, the stack $\mathcal{X}$ can be recovered as a weighted blow-up along $\mathcal{Y}\subset \mathcal{Z}$ with exceptional divisor $\mathcal{E}$, and $\mathcal{Z}$ is a pushout in the category of algebraic stacks. As an application, we show that the moduli stack $\overline{\mathscr{M}}_{1,n}$ of stable $n$-pointed genus one curves is a weighted blow-up of the stack of pseudo-stable curves. Along the way we also prove a reconstruction result for smooth Deligne-Mumford stacks that is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_15076 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A criterion for smooth weighted blow-downs Arena, Veronica Di Lorenzo, Andrea Inchiostro, Giovanni Mathur, Siddharth Obinna, Stephen Pernice, Michele Algebraic Geometry 14D23, 14E05 We establish a criterion for determining when a smooth Deligne-Mumford stack is a weighted blow-up. More precisely, given a smooth Deligne-Mumford stack $\mathcal{X}$ and a Cartier divisor $\mathcal{E} \subset \mathcal{X}$ such that (1) $\mathcal{E}$ is a weighted projective bundle over a smooth Deligne-Mumford stack $\mathcal{Y}$ and (2) for every $y\in\mathcal{Y}$ we have $\mathcal{O}_{\mathcal{X}}(\mathcal{E})|_{\mathcal{E}_y}\simeq \mathcal{O}_{\mathcal{E}_y}(-1)$, then there exists a contraction $\mathcal{X}\to\mathcal{Z}$ to a smooth Deligne-Mumford stack $\mathcal{Z}$. Moreover, the stack $\mathcal{X}$ can be recovered as a weighted blow-up along $\mathcal{Y}\subset \mathcal{Z}$ with exceptional divisor $\mathcal{E}$, and $\mathcal{Z}$ is a pushout in the category of algebraic stacks. As an application, we show that the moduli stack $\overline{\mathscr{M}}_{1,n}$ of stable $n$-pointed genus one curves is a weighted blow-up of the stack of pseudo-stable curves. Along the way we also prove a reconstruction result for smooth Deligne-Mumford stacks that is of independent interest. |
| title | A criterion for smooth weighted blow-downs |
| topic | Algebraic Geometry 14D23, 14E05 |
| url | https://arxiv.org/abs/2310.15076 |