13 Solvable Groups, Free Divisors and Nonisolated MatrixSingularities II: Vanishing Topology
Bibliographic record
Abstract
In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices.We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for representations of solvable groups.Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors E n , and we give an inductive procedure for computing the vanishing topology of the matrix singularities.The inductive procedure we use is an extension of that introduced by Lê-Greuel for computing the Milnor number of an ICIS.Instead of linear subspaces, we use free divisors arising from the geometric configuration and which correspond to subgroups of the solvable groups.Here the vanishing topology involves a singular version of the Milnor fiber; however, it still has the good connectivity properties and is homotopy equivalent to a bouquet of spheres, whose number is called the singular Milnor number.We give formulas for this singular Milnor number in terms of singular Milnor numbers of various free divisors on smooth subspaces, which can be computed as lengths of determinantal modules.In addition to being applied to symmetric, general and skew-symmetric matrix singularities, the results are also applied to Cohen-Macaulay singularities defined as 2 3 matrix singularities.We compute the Milnor number of isolated Cohen-Macaulay surface singularities of this type in C 4 and the difference of Betti numbers of Milnor fibers for isolated Cohen-Macaulay 3-fold singularities of this type in C 5 .
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 source (direct Gemma or distilled Codex), 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".