Seshadri constants on $\mathbb {P}^1\times \mathbb {P}^1$ and applications to the symplectic packing problem
Bibliographic record
Abstract
Abstract In this paper we compute the r -point Seshadri constant on $\mathbb {P}^1\times \mathbb {P}^1$ for those line bundles where the answer might be expected to be governed by $(-1)$ -curves. As a consequence we obtain explicit formulas for the symplectic packing problem for $\mathbb {P}^1\times \mathbb {P}^1$ . Some exact values for the r -point Seshadri constant outside the region governed by Mori’s cone theorem are also given. These latter results use a useful new “reflection method”. In the analysis there is a striking difference between the cases when r is odd and when r is even. When r is even the problem admits an infinite order automorphism, and there are infinitely many $(-1)$ -curves to consider. In contrast, when r is odd only a finite number (usually four) types of $(-1)$ -curves are relevant to our answer.
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.002 |
| 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".