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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.01859 |
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| _version_ | 1866909938449645568 |
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| author | Brais, Maxim Jean-Louis |
| author_facet | Brais, Maxim Jean-Louis |
| contents | In 2019, Abramovich--Temkin--Wlodarczyk and McQuillan used weighted blow-ups to obtain very fast and functorial algorithms for resolution of singularities in characteristic zero. Recently, Abramovich--Quek--Schober simplified the construction of the centre of blow-up introduced by Abramovich--Temkin--Wlodarczyk in the case of plane curves by using the Newton graph of the defining function. Their work follows the line of Schober's previous polyhedral analysis of the Bierstone--Milman invariant. In this paper, we extend their graphical approach to varieties of arbitrary (co)dimension in characteristic zero. This yields a factorial reduction in complexity in comparison with Abramovich--Temkin--Wlodarczyk, as previously achieved by Wlodarczyk. Our approach builds on the formalism of weighted blow-ups via filtrations of ideals developed and used by Loizides--Meinrenken, Quek--Rydh and Wlodarczyk, and on its interplay with systems of parameters. All constructions and proofs -- including that of resolution of varieties by Deligne--Mumford stacks -- are self-contained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01859 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Streamlining resolution of singularities with weighted blow-ups Brais, Maxim Jean-Louis Algebraic Geometry In 2019, Abramovich--Temkin--Wlodarczyk and McQuillan used weighted blow-ups to obtain very fast and functorial algorithms for resolution of singularities in characteristic zero. Recently, Abramovich--Quek--Schober simplified the construction of the centre of blow-up introduced by Abramovich--Temkin--Wlodarczyk in the case of plane curves by using the Newton graph of the defining function. Their work follows the line of Schober's previous polyhedral analysis of the Bierstone--Milman invariant. In this paper, we extend their graphical approach to varieties of arbitrary (co)dimension in characteristic zero. This yields a factorial reduction in complexity in comparison with Abramovich--Temkin--Wlodarczyk, as previously achieved by Wlodarczyk. Our approach builds on the formalism of weighted blow-ups via filtrations of ideals developed and used by Loizides--Meinrenken, Quek--Rydh and Wlodarczyk, and on its interplay with systems of parameters. All constructions and proofs -- including that of resolution of varieties by Deligne--Mumford stacks -- are self-contained. |
| title | Streamlining resolution of singularities with weighted blow-ups |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2512.01859 |