Logarithmic resolution via multi-weighted blow-ups
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910258649104384 |
|---|---|
| author | Abramovich, Dan Quek, Ming Hao |
| author_facet | Abramovich, Dan Quek, Ming Hao |
| contents | We first introduce and study the notion of multi-weighted blow-ups, which is later used to systematically construct an explicit yet efficient algorithm for functorial logarithmic resolution in characteristic zero, in the sense of Hironaka. Specifically, for a singular, reduced closed subscheme $X$ of a smooth scheme $Y$ over a field of characteristic zero, we resolve the singularities of $X$ by taking proper transforms $X_i \subset Y_i$ along a sequence of multi-weighted blow-ups $Y_N \to Y_{N-1} \to \dotsb \to Y_0 = Y$ which satisfies the following properties: (i) the $Y_i$ are smooth Artin stacks with simple normal crossing exceptional loci; (ii) at each step we always blow up the worst singular locus of $X_i$, and witness on $X_{i+1}$ an immediate improvement in singularities; (iii) and finally, the singular locus of $X$ is transformed into a simple normal crossing divisor on $X_N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_06361 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Logarithmic resolution via multi-weighted blow-ups Abramovich, Dan Quek, Ming Hao Algebraic Geometry 14E15 (Primary), 14A20, 14M25 (Secondary) We first introduce and study the notion of multi-weighted blow-ups, which is later used to systematically construct an explicit yet efficient algorithm for functorial logarithmic resolution in characteristic zero, in the sense of Hironaka. Specifically, for a singular, reduced closed subscheme $X$ of a smooth scheme $Y$ over a field of characteristic zero, we resolve the singularities of $X$ by taking proper transforms $X_i \subset Y_i$ along a sequence of multi-weighted blow-ups $Y_N \to Y_{N-1} \to \dotsb \to Y_0 = Y$ which satisfies the following properties: (i) the $Y_i$ are smooth Artin stacks with simple normal crossing exceptional loci; (ii) at each step we always blow up the worst singular locus of $X_i$, and witness on $X_{i+1}$ an immediate improvement in singularities; (iii) and finally, the singular locus of $X$ is transformed into a simple normal crossing divisor on $X_N$. |
| title | Logarithmic resolution via multi-weighted blow-ups |
| topic | Algebraic Geometry 14E15 (Primary), 14A20, 14M25 (Secondary) |
| url | https://arxiv.org/abs/2112.06361 |