Resolution Except for the Normal-Crossing Locus and Galois actions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910104482217984 |
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| author | Włodarczyk, Jarosław |
| author_facet | Włodarczyk, Jarosław |
| contents | In characteristic zero, we construct a canonical, functorial resolution algorithm by weighted blow-ups that strictly preserves the normal crossings (nc) locus, effectively answering Kollar's problem. Operating in full generality, our approach handles both reduced and non-reduced nc singularities alongside the simple normal crossings (snc) exceptional divisor setup, terminating with a normal crossings Deligne-Mumford stack.
The resolution is governed by two fundamental geometric properties: the openness of the nc locus and the topological rigidity of canonical maximal admissible weighted centers (the Center Theorem). We establish these via a direct Galois-theoretic analysis of splitting forms. By viewing general nc singularities as quotients of 'etale-local snc singularities by finite Galois groups permuting their branches, we reveal the intrinsic necessity of weighted blow-ups and stack structures.
Building on the weighted blow-up framework of Abramovich, Temkin, and Wlodarczyk and our logarithmic refinement, this structural mechanism yields a robust resolution algorithm. Applications include a canonical compactification of nc Deligne-Mumford stacks and a functorial nc-preserving resolution of subvarieties and stacks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_14266 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Resolution Except for the Normal-Crossing Locus and Galois actions Włodarczyk, Jarosław Algebraic Geometry 14E15, 14E18, 14D06, 14A20, 14B05 In characteristic zero, we construct a canonical, functorial resolution algorithm by weighted blow-ups that strictly preserves the normal crossings (nc) locus, effectively answering Kollar's problem. Operating in full generality, our approach handles both reduced and non-reduced nc singularities alongside the simple normal crossings (snc) exceptional divisor setup, terminating with a normal crossings Deligne-Mumford stack. The resolution is governed by two fundamental geometric properties: the openness of the nc locus and the topological rigidity of canonical maximal admissible weighted centers (the Center Theorem). We establish these via a direct Galois-theoretic analysis of splitting forms. By viewing general nc singularities as quotients of 'etale-local snc singularities by finite Galois groups permuting their branches, we reveal the intrinsic necessity of weighted blow-ups and stack structures. Building on the weighted blow-up framework of Abramovich, Temkin, and Wlodarczyk and our logarithmic refinement, this structural mechanism yields a robust resolution algorithm. Applications include a canonical compactification of nc Deligne-Mumford stacks and a functorial nc-preserving resolution of subvarieties and stacks. |
| title | Resolution Except for the Normal-Crossing Locus and Galois actions |
| topic | Algebraic Geometry 14E15, 14E18, 14D06, 14A20, 14B05 |
| url | https://arxiv.org/abs/2602.14266 |