Bibliographic record
Abstract
Let φ : X → S be a morphism between smooth complex analytic spaces and let f = 0 define a free divisor on S. We prove that if the deformation space T X / S 1 of φ is a Cohen–Macaulay O X -module of codimension 2, and all of the logarithmic vector fields for f = 0 lift via φ, then f ∘ φ = 0 defines a free divisor on X; this is generalized in several directions. Among applications we recover a result of Mond–van Straten, generalize a construction of Buchweitz–Conca, and show that a map φ : C n + 1 → C n with critical set of codimension 2 has a T X / S 1 with the desired properties. Finally, if X is a representation of a reductive complex algebraic group G and φ is the algebraic quotient X → S = X / / G with X / / G smooth, then we describe sufficient conditions for T X / S 1 to be Cohen–Macaulay of codimension 2. In one such case, a free divisor on C n + 1 lifts under the operation of ‘castling’ to a free divisor on C n ( n + 1 ) , partially generalizing work of Granger–Mond–Schulze on linear free divisors. We give several other examples of such representations.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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".