Constructing geometric realizations of birational maps between Mori Dream Spaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913768106098688 |
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| author | Barban, Lorenzo Occhetta, Gianluca Conde, Luis E. Sol á |
| author_facet | Barban, Lorenzo Occhetta, Gianluca Conde, Luis E. Sol á |
| contents | We construct geometric realizations -- projective algebraic versions of cobordisms -- for birational maps between Mori Dream Spaces. We show that these geometric realizations are Mori Dream Spaces, as well, and that they can be constructed so that they induce factorizations of the original birational maps as compositions of wall-crossings. In the case of toric birational maps between normal $\mathbb{Q}$-factorial, projective toric varieties, we provide several SageMath functions to work with $\mathbb{C}^*$-actions and birational geometry; in particular we show how to explicitly construct a moment polytope of a toric geometric realization. Moreover, by embedding Mori Dream Spaces in toric varieties, we obtain geometric realizations of birational maps of Mori Dream Spaces as restrictions of toric geometric realizations. We also provide examples and discuss when a geometric realization is Fano. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23790 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructing geometric realizations of birational maps between Mori Dream Spaces Barban, Lorenzo Occhetta, Gianluca Conde, Luis E. Sol á Algebraic Geometry Primary 14E30, Secondary 14L30, 14L24, 14M25 We construct geometric realizations -- projective algebraic versions of cobordisms -- for birational maps between Mori Dream Spaces. We show that these geometric realizations are Mori Dream Spaces, as well, and that they can be constructed so that they induce factorizations of the original birational maps as compositions of wall-crossings. In the case of toric birational maps between normal $\mathbb{Q}$-factorial, projective toric varieties, we provide several SageMath functions to work with $\mathbb{C}^*$-actions and birational geometry; in particular we show how to explicitly construct a moment polytope of a toric geometric realization. Moreover, by embedding Mori Dream Spaces in toric varieties, we obtain geometric realizations of birational maps of Mori Dream Spaces as restrictions of toric geometric realizations. We also provide examples and discuss when a geometric realization is Fano. |
| title | Constructing geometric realizations of birational maps between Mori Dream Spaces |
| topic | Algebraic Geometry Primary 14E30, Secondary 14L30, 14L24, 14M25 |
| url | https://arxiv.org/abs/2503.23790 |