Double sections, dominating maps, and the Jacobian fibration
Bibliographic record
Abstract
We give two parametrized versions of the uniformization theorem of a noncompact, non-hyperbolic Riemann surface using different but complementary methods. The first constructs the uniformizing maps directly in terms of coordinates via classical complex analysis and provides a canonical form for the double sections of a conic bundle over a noncompact complex curve. The second version, which is coordinate independent, works over any complex curve and is obtained by extending Kodaira's theory of the Jacobian fibration to a family of singular algebraic curves constructed via algebraic geometry. Then, using the results obtained with the Jacobian fibration, we give two equivalent conditions for a complex analytic surface nonhyperbolically fibered over a complex curve to be holomorphically dominable by [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]: We show that this dominability is equivalent to the apparently weaker condition of the existence of a Zariski dense image of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] and equivalent to the quasiprojectivity of the base curve together with the nonnegativity of the orbifold Euler characteristic. We discuss also the sharpness of our result in various contexts as well as the lack of connection to the fundamental group.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.000 | 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".