Bibliographic record
Abstract
Given an algebraic subvariety X of a multiplicative torus A over , we study the intersection of with the union A[m] of all algebraic subgroups of codimension . This is related to conjectures stated by Zilber and Pink and follows previous work by Bombieri, Masser and Zannier. Our main result can be stated as follows: if X is a nondegenerate subvariety of A, then for any subgroup of finite rank Γ of , the points lying in a height-theoretic cone with nonzero radius around ΓA[m] are not dense in X. The case Γ={1} was proved by Habegger in 2009 as a corollary to his bounded height theorem. We show here that the general case follows from a new Vojta-type height inequality on Xm. The proof of this inequality relies on diophantine approximation techniques coming from previous work by Vojta, Faltings, Bombieri, … , Rémond. It shows that the height inequality can be deduced from a lower bound on some intersection numbers coming from a family of blow-ups of a given compactification of Xm. Our main task is to establish this lower bound. For that purpose, we use homogeneity and positivity properties of the intersection numbers. They are proved here in an analytic setting, using integrals of Chern-like forms over suitably chosen desingularizations.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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".