Fat Points in ℙ<sup>1</sup>× ℙ<sup>1</sup>and Their Hilbert Functions
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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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".