Positivity criteria for log canonical divisors and hyperbolicity
Bibliographic record
Abstract
Abstract LetXbe a complex projective variety andDa reduced divisor onX. Under mild conditions on the singularities of (X,D) $(X,D)$ , which includes the case of smoothXwith simple normal crossingD, and by running the minimal model program, we obtain by induction on dimension via adjunction geometric criteria guaranteeing various positivity conditions for KX+D $K_{X}+D$ . Our geometric criterion for KX+D $K_{X}+D$ to be numerically effective yields also a geometric version of the cone theorem and a criterion for KX+D $K_{X}+D$ to be pseudo-effective with mild hypothesis onD. We also obtain, assuming the abundance conjecture and the existence of rational curves on Calabi–Yau manifolds, an optimal geometric sharpening of the Nakai–Moishezon criterion for the ampleness of a divisor of the form KX+D $K_{X}+D$ , a criterion verified under a canonical hyperbolicity assumption on (X,D) $(X,D)$ . Without these conjectures, we verify this ampleness criterion with mild assumptions onD, being none in dimension two and D≠0 $D\neq 0$ in dimension three.
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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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.013 | 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".