On hypersurfaces containing projective varieties
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Bibliographic record
Abstract
Abstract Classical Castelnuovo's lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension c is at most <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfenced separators="" open="(" close=")"> <m:mfrac linethickness="0pt"> <m:mrow> <m:mi>c</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mn>2</m:mn> </m:mfrac> </m:mfenced> </m:math> ${{\binom{c+1}{2}}}$ and the equality is attained if and only if the variety is of minimal degree. Also a generalization of Castelnuovo's lemma by G. Fano implies that the next case occurs if and only if the variety is a del Pezzo variety. For curve case, these results are extended to equations of arbitrary degree respectively by J. Harris and S. L'vovsky. This paper is intended to extend these results to arbitrary dimensional varieties and to the next cases.
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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 | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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