Higher-dimensional Losev-Manin spaces and their geometry
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912999518765056 |
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| author | Gallardo, Patricio González-Anaya, Javier González, José Luis Routis, Evangelos |
| author_facet | Gallardo, Patricio González-Anaya, Javier González, José Luis Routis, Evangelos |
| contents | The classical Losev-Manin space is a toric compactification of the moduli space of $n$ points in the affine line modulo translation and scaling. Motivated by this, we study its higher-dimensional toric counterparts, which compactify the moduli space of $n$ distinct labeled points in affine space modulo translation and scaling. We show that these moduli spaces are a fibration over a product of projective spaces -- with fibers isomorphic to the Losev-Manin space -- and that they are isomorphic to the normalization of a Chow quotient. Moreover, we present a criterion to decide whether the blow-up of a toric variety along the closure of a subtorus is a Mori dream space. As an application, we demonstrate that a related generalization of the moduli space of pointed rational curves constructed by Chen, Gibney, and Krashen is not a Mori dream space when the number of points is at least nine, regardless of the dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07911 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher-dimensional Losev-Manin spaces and their geometry Gallardo, Patricio González-Anaya, Javier González, José Luis Routis, Evangelos Algebraic Geometry 14C05, 14E30, 14D06, 14D20 (Primary) 14M25, 14L30 (Secondary) The classical Losev-Manin space is a toric compactification of the moduli space of $n$ points in the affine line modulo translation and scaling. Motivated by this, we study its higher-dimensional toric counterparts, which compactify the moduli space of $n$ distinct labeled points in affine space modulo translation and scaling. We show that these moduli spaces are a fibration over a product of projective spaces -- with fibers isomorphic to the Losev-Manin space -- and that they are isomorphic to the normalization of a Chow quotient. Moreover, we present a criterion to decide whether the blow-up of a toric variety along the closure of a subtorus is a Mori dream space. As an application, we demonstrate that a related generalization of the moduli space of pointed rational curves constructed by Chen, Gibney, and Krashen is not a Mori dream space when the number of points is at least nine, regardless of the dimension. |
| title | Higher-dimensional Losev-Manin spaces and their geometry |
| topic | Algebraic Geometry 14C05, 14E30, 14D06, 14D20 (Primary) 14M25, 14L30 (Secondary) |
| url | https://arxiv.org/abs/2308.07911 |