A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves
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Abstract
We determine an expression <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi Subscript g Superscript s Baseline left-parenthesis gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi> ξ </mml:mi> <mml:mi>g</mml:mi> <mml:mi>s</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> γ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\xi ^s_g(\gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the virtual Euler characteristics of the moduli spaces of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -pointed real <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis gamma equals 1 slash 2"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> γ </mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">(\gamma =1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) and complex ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma equals 1"> <mml:semantics> <mml:mrow> <mml:mi> γ </mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\gamma =1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) algebraic curves. In particular, for the space of real curves of genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with a fixed point free involution, we find that the Euler characteristic is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis negative 2 right-parenthesis Superscript s minus 1 Baseline left-parenthesis 1 minus 2 Superscript g minus 1 Baseline right-parenthesis left-parenthesis g plus s minus 2 right-parenthesis factorial upper B Subscript g Baseline slash g factorial"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>+</mml:mo> <mml:mi>s</mml:mi> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>!</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>g</mml:mi> <mml:mo>!</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(-2)^{s-1}(1-2^{g-1})(g+s-2)!B_g/g!</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript g"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">B_g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis negative 1 right-parenthesis Superscript s Baseline left-parenthesis g plus s minus 2 right-parenthesis factorial upper B Subscript g plus 1 Baseline slash left-parenthesis g plus 1 right-parenthesis left-parenthesis g minus 1 right-parenthesis factorial"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>+</mml:mo> <mml:mi>s</mml:mi> <mml:mo>
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| 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 it