Bibliographic record
Abstract
As is well-known, there exist nonconstant holomorphic maps from the plane into the Riemann sphere $\PP^1$ minus two points, the simplest example of which is an explicit realization of the uniformization map given by applying the exponential map and then composing with a Mobius transformation taking 0 and $\infty$ to the two given punctures. Likewise, we can map the plane into the sphere minus one point by simply applying directly a Mobius transformation taking $\infty$ to this puncture. In this paper we prove two parametrized versions of this uniformization result using different methods. We first present a slightly easier version using complex analysis, which allows us to construct the uniformizing maps more or less explicitly in terms of the initial coordinates given. Then we give the more general and coordinate invariant version, which we prove by extending Kodaira's theory of the Jacobian fibration to a family of singular curves constructed via algebraic geometry. Finally, using the results obtained with the Jacobian fibration, we obtain two equivalent conditions for an complex surface nonhyperbolically fibered over a complex curve to be holomorphically dominable by $\CC^2$. In theorem \ref{thm:surf} we show that this dominability is equivalent to the apparently weaker condition of the existence of a Zariski dense image of $\CC$ and equivalent to the quasiprojectivity of the base curve together with the nonnegativity of the orbifold Euler characteristic.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".