Multi-scale differentials and wonderful models
Bibliographic record
Abstract
We study the relationships between several varieties parametrizing marked curves with differentials in the literature. More precisely, we prove that the space B n \mathcal {B}_n of multi-scale differentials of genus 0 0 with n + 1 n+1 marked points of orders ( 0 , … , 0 , − 2 ) (0,\ldots , 0, -2) , as introduced by Matt Bainbridge, Dawei Chen, Quentin Gendron, Samuel Grushevsky, and Martin Möller [ The moduli space of multi-scale differentials , 2022], is a wonderful variety in the sense of work by Tom Braden, June Huh, Jacob P. Matherne, Nicholas Proudfoot, and Botong Wang [Adv. Math. 409 (2022), Paper No. 108646, 49]. This shows that the Chow ring of B n \mathcal {B}_n is generated by the classes of a collection of smooth boundary divisors with normal crossings subject to simple and explicit linear and quadratic relations. Furthermore, we realize B n \mathcal {B}_n as a subvariety of the space A n \mathcal {A}_n of multi-scale lines defined by Daniel Halpern-Leistner and Antonios-Alexandros Robotis [Preprint, arXiv: 2501.00710 , 2025] and prove that B n \mathcal {B}_n can be realized as the normalized Chow quotient of A n \mathcal {A}_n by a natural C ∗ \mathbb {C}^* -action.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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".