Fat Points in ℙ<sup>1</sup>× ℙ<sup>1</sup>and Their Hilbert Functions
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Bibliographic record
Abstract
Abstract We study the Hilbert functions of fat points in ℙ 1 × ℙ 1 . If Z ⊆ ℙ 1 × ℙ 1 is an arbitrary fat point scheme, then it can be shown that for every i and j the values of the Hilbert function H Z ( l , j ) and H Z ( i , l ) eventually become constant for l ≫ 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ℙ 1 × ℙ 1 . This enables us to compute all but a finite number values of H Z without using the coordinates of points. We also characterize the ACM fat point schemes using our description of the eventual behaviour. In fact, in the case that Z ⊆ ℙ 1 × ℙ 1 is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.
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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 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 it