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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2411.16141 |
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| _version_ | 1866916631666491392 |
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| author | Di Lorenzo, Andrea Inchiostro, Giovanni |
| author_facet | Di Lorenzo, Andrea Inchiostro, Giovanni |
| contents | We compactify the moduli stack of maps from curves to certain quotient stacks $\mathcal{X}=[W/G]$ with a projective good moduli space, extending previous results from quasimap theory. For doing so, we introduce a new birational transformation for algebraic stacks, the extended weighted blow-up, to prove that any algebraic stack with a properly stable point can be enlarged so that it contains an open substack which is proper and Deligne-Mumford. As a first application, we use our main theorem to construct a compact moduli stack for certain fibered log-Calabi-Yau pairs. We further apply our result to construct a compactification of the space of maps to $\mathcal{X}$ when $\mathcal{X}$ is respectively: a quotient by a torus of a proper Deligne-Mumford stack; a GIT compactification of the stack of binary forms of degree $2n$; a GIT compactification of the stack of $2n$-marked smooth rational curves, and a GIT compactification of the stack of smooth plane cubics. In the appendix, we give a criterion for when a morphism of algebraic stacks is an extended weighted blow-up, and we use it in order to give a modular proof of a conjecture of Hassett on weighted pointed rational curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16141 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable maps to quotient stacks with a properly stable point Di Lorenzo, Andrea Inchiostro, Giovanni Algebraic Geometry We compactify the moduli stack of maps from curves to certain quotient stacks $\mathcal{X}=[W/G]$ with a projective good moduli space, extending previous results from quasimap theory. For doing so, we introduce a new birational transformation for algebraic stacks, the extended weighted blow-up, to prove that any algebraic stack with a properly stable point can be enlarged so that it contains an open substack which is proper and Deligne-Mumford. As a first application, we use our main theorem to construct a compact moduli stack for certain fibered log-Calabi-Yau pairs. We further apply our result to construct a compactification of the space of maps to $\mathcal{X}$ when $\mathcal{X}$ is respectively: a quotient by a torus of a proper Deligne-Mumford stack; a GIT compactification of the stack of binary forms of degree $2n$; a GIT compactification of the stack of $2n$-marked smooth rational curves, and a GIT compactification of the stack of smooth plane cubics. In the appendix, we give a criterion for when a morphism of algebraic stacks is an extended weighted blow-up, and we use it in order to give a modular proof of a conjecture of Hassett on weighted pointed rational curves. |
| title | Stable maps to quotient stacks with a properly stable point |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2411.16141 |