Bibliographic record
Abstract
We construct curve counting invariants for a Calabi-Yau threefold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> equipped with a dominant birational morphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi colon upper Y right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi> π </mml:mi> <mml:mo>:</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi :Y \to X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Our invariants generalize the stable pair invariants of Pandharipande and Thomas which occur for the case when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi colon upper Y right-arrow upper Y"> <mml:semantics> <mml:mrow> <mml:mi> π </mml:mi> <mml:mo>:</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi :Y\to Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the identity. Our main result is a PT/DT-type formula relating the partition function of our invariants to the Donaldson-Thomas partition function in the case when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a crepant resolution of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the coarse space of a Calabi-Yau orbifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying the hard Lefschetz condition. In this case, our partition function is equal to the Pandharipande-Thomas partition function of the orbifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Our methods include defining a new notion of stability for sheaves which depends on the morphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Our notion generalizes slope stability which is recovered in the case where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the identity on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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 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".