The Abel–Jacobi isomorphism for one-cycles on Kirwan's log resolution of the moduli space𝒮𝒰<sub> <i>C</i> </sub>(2,𝒪<sub> <i>C</i> </sub>)
Bibliographic record
Abstract
Abstract In this paper, we consider the moduli space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝒮</m:mi> <m:mspace width="-1.0pt" /> <m:msub> <m:mi>𝒰</m:mi> <m:mi>C</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>𝒪</m:mi> <m:mi>C</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $\mathcal {S\hspace{-1.0pt}U}_C(r,\mathcal {O}_C)$ of rank r semistable vector bundles with trivial determinant on a smooth projective curve C of genus g . For r = 2, F. Kirwan constructed a smooth log resolution <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mover> <m:mi>X</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mo>→</m:mo> <m:mi>𝒮</m:mi> <m:mspace width="-1.0pt" /> <m:msub> <m:mi>𝒰</m:mi> <m:mi>C</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:msub> <m:mi>𝒪</m:mi> <m:mi>C</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $\overline{X}\rightarrow \mathcal {S\hspace{-1.0pt}U}_C(2,\mathcal {O}_C)$ . Based on earlier work of M. Kerr and J. Lewis, Lewis explains in the Appendix the notion of a relative Chow group (w.r.t. the normal crossing divisor), and a subsequent Abel–Jacobi map on the relative Chow group of null-homologous one-cycles (tensored with ℚ). This map takes values in the intermediate Jacobian of the compactly supported cohomology of the stable locus. We show that this is an isomorphism and since the intermediate Jacobian is identified with the Jacobian <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo form="prefix">Jac</m:mo> <m:mo>(</m:mo> <m:mi>C</m:mi> <m:mo>)</m:mo> <m:mo>⊗</m:mo> <m:mi>ℚ</m:mi> </m:mrow> </m:math> $\operatorname{Jac}(C)\otimes \mathbb {Q}$ , this can be thought of as a weak-representability result for open smooth varieties. A hard Lefschetz theorem is also proved for the odd degree bottom weight cohomology of the moduli space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝒮</m:mi> <m:mspace width="-1.0pt" /> <m:msubsup> <m:mi>𝒰</m:mi> <m:mi>C</m:mi> <m:mi>s</m:mi> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:msub> <m:mi>𝒪</m:mi> <m:mi>C</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $\mathcal {S\hspace{-1.0pt}U}_C^s(2,\mathcal {O}_C)$ . When r ≥ 2, we compute the codimension two rational Chow groups of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝒮</m:mi> <m:mspace width="-1.0pt" /> <m:msub> <m:mi>𝒰</m:mi> <m:mi>C</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>𝒪</m:mi> <m:mi>C</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $\mathcal {S\hspace{-1.0pt}U}_C(r,\mathcal {O}_C)$ .
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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 | 0.003 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.003 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| 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".