Streamlining resolution of singularities with weighted blow-ups
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Bibliographic record
Abstract
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.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it