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Motivic Euler characteristics of moduli spaces of curves
Algebraic Geometry · Submitted 7 Oct 2026
abstract compiled · 2026-10-08 · no language-model review
The authors' main claim We compute the motivic Euler characteristic for $9$ more pairs, leaving only $6$ pairs where the motivic Euler characteristic is polynomial but unknown.
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The abstract, sentence by sentence
A combination of recent results of Canning--Larson--Payne--Willwacher and Payne--Willwacher is that the motivic Euler characteristic of the moduli space $M_{g,n}$ of nonsingular curves with $n$ markings is polynomial in the Tate motive if and only if $g=0$ or $3g+2n<25$. In the cases when $g\geq 1$ and $3g+2n<25$, there are $32$ pairs $(g,n)$ where the motivic Euler characteristic has been computed. We compute the motivic Euler characteristic for $9$ more pairs, leaving only $6$ pairs where the motivic Euler characteristic is polynomial but unknown. In genus $4$, we moreover compute the answer when $n=7$, i.e.~the first non-polynomial case.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A combination of recent results of Canning--Larson--Payne--Willwacher and Payne--Willwacher is that the motivic Euler characteristic of the moduli space $M_{g,n}$ of nonsingular curves with $n$ markings is polynomial in the Tate motive if and only if $g=0$ or $3g+2n<25$. | background 0.99 | 0.04 | 0.22 | asserted 1.00 |
| s1 | In the cases when $g\geq 1$ and $3g+2n<25$, there are $32$ pairs $(g,n)$ where the motivic Euler characteristic has been computed. | background 0.73 | 0.15 | 0.06 | asserted 1.00 |
| s2 | We compute the motivic Euler characteristic for $9$ more pairs, leaving only $6$ pairs where the motivic Euler characteristic is polynomial but unknown. | contribution 0.82 | 0.96 | 0.55 | asserted 1.00 |
| s3 | In genus $4$, we moreover compute the answer when $n=7$, i.e.~the first non-polynomial case. | result 0.81 | 0.90 | 0.12 | asserted 1.00 |
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The paper
Original abstract
A combination of recent results of Canning--Larson--Payne--Willwacher and Payne--Willwacher is that the motivic Euler characteristic of the moduli space $M_{g,n}$ of nonsingular curves with $n$ markings is polynomial in the Tate motive if and only if $g=0$ or $3g+2n<25$. In the cases when $g\geq 1$ and $3g+2n<25$, there are $32$ pairs $(g,n)$ where the motivic Euler characteristic has been computed. We compute the motivic Euler characteristic for $9$ more pairs, leaving only $6$ pairs where the motivic Euler characteristic is polynomial but unknown. In genus $4$, we moreover compute the answer when $n=7$, i.e.~the first non-polynomial case.
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