Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces
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| Format: | Preprint |
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2021
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| _version_ | 1866910265127206912 |
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| author | Bruce, Juliette Heller, Lauren Cranton Sayrafi, Mahrud |
| author_facet | Bruce, Juliette Heller, Lauren Cranton Sayrafi, Mahrud |
| contents | We explore the relationship between multigraded Castelnuovo--Mumford regularity, truncations, Betti numbers, and virtual resolutions on a product of projective spaces $X$. After proving a uniqueness theorem for certain virtual resolutions, we show that the multigraded regularity region of a module $M$ is determined by the minimal graded free resolutions of the truncations $M_{\geq\mathbf d}$ for $\mathbf d\in\operatorname{Pic} X$. Further, by relating the minimal graded free resolutions of $M$ and $M_{\geq\mathbf d}$ we provide a new bound on multigraded regularity of $M$ in terms of its Betti numbers. Using this characterization of regularity and this bound we also compute the multigraded Castelnuovo--Mumford regularity for a wide class of complete intersections in products of projective spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_10705 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces Bruce, Juliette Heller, Lauren Cranton Sayrafi, Mahrud Commutative Algebra Algebraic Geometry 13D02, 14M25 We explore the relationship between multigraded Castelnuovo--Mumford regularity, truncations, Betti numbers, and virtual resolutions on a product of projective spaces $X$. After proving a uniqueness theorem for certain virtual resolutions, we show that the multigraded regularity region of a module $M$ is determined by the minimal graded free resolutions of the truncations $M_{\geq\mathbf d}$ for $\mathbf d\in\operatorname{Pic} X$. Further, by relating the minimal graded free resolutions of $M$ and $M_{\geq\mathbf d}$ we provide a new bound on multigraded regularity of $M$ in terms of its Betti numbers. Using this characterization of regularity and this bound we also compute the multigraded Castelnuovo--Mumford regularity for a wide class of complete intersections in products of projective spaces. |
| title | Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces |
| topic | Commutative Algebra Algebraic Geometry 13D02, 14M25 |
| url | https://arxiv.org/abs/2110.10705 |