Billey-Postnikov decompositions and the fibre bundle structure of Schubert\n varieties
Bibliographic record
Abstract
A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and\n only if it is an iterated fibre bundle of Grassmannians. We extend this theorem to\n arbitrary finite type, showing that a Schubert variety in a generalized flag variety is\n rationally smooth if and only if it is an iterated fibre bundle of rationally smooth\n Grassmannian Schubert varieties. The proof depends on deep combinatorial results of\n Billey-Postnikov on Weyl groups. We determine all smooth and rationally smooth Grassmannian\n Schubert varieties, and give a new proof of Peterson's theorem that all simply-laced\n rationally smooth Schubert varieties are smooth. Taken together, our results give a fairly\n complete geometric description of smooth and rationally smooth Schubert varieties using\n primarily combinatorial methods. We also give some partial results for Schubert varieties\n in Kac-Moody flag varieties. In particular, we show that rationally smooth Schubert\n varieties of affine type A are also iterated fibre bundles of rationally smooth\n Grassmannian Schubert varieties. As a consequence, we finish a conjecture of Billey-Crites\n that a Schubert variety in affine type A is smooth if and only if the corresponding affine\n permutation avoids the patterns 4231 and 3412.
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How this classification was reachedexpand
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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".