The fifteenth cohomology group of moduli spaces of stable curves
Abstract
The new framework of semi-tautological extensions has allowed for explicit understanding of low-degree odd cohomology groups of moduli spaces of stable curves. Let S*ω(ℳ̅g,n) ⊂ H*(ℳ̅g,n) denote the semi-tautological extension generated by H11(ℳ̅1,11), that is, the smallest system of subrings of H*(ℳ̅g,n) containing H11(ℳ̅1,11) and the tautological subrings, which is also closed under pullbacks along forgetful maps and pushforwards along gluing maps. Recently, it has been shown that H11(ℳ̅g,n) = S11ω(ℳ̅g,n) and H13(ℳ̅g,n) = S13ω(ℳ̅g,n) for all g and n. Meanwhile, H15(ℳ̅1,n) ≠ Sω15(ℳ̅1,n) for all n ≥ 15. Nevertheless, we prove that H15(ℳ̅2,n) = S15ω(ℳ̅2,n) for all n. Moreover, we determine all relations in Sω15(ℳ̅g,n) for all g ≥ 2 and give explicit bases for this vector space in terms of decorated graphs. Conditional upon the conjectured dimension of H15(ℳ̅3,14), we also prove that H15(ℳ̅3,n) = S15ω(ℳ̅3,n) for all n. As an application, we prove H*(ℳ̅g,n) = RH*(ℳ̅g,n) for several new small pairs (g,n).
Related
- Stable rational cohomology splitting of moduli of branched covers of curves
- A Murphy's law for supports of forgetful maps between moduli spaces of stable pointed curves
- An action of the affine Yangian Yt1,t2(gl̂1) on the cohomology of the moduli space of stable sheaves on surfaces
- Stability of representation cohomology sheaves on moduli spaces of bundles
- Cetraro lectures on integrable systems and moduli spaces of curves
BibTeX
@misc{bradford2026fifteenth,
title = {{The fifteenth cohomology group of moduli spaces of stable curves}},
author = {Amy Bradford and Aitor Iribar López and Andreas Kretschmer and Hannah Larson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14377},
eprint = {2610.14377},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14377},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The fifteenth cohomology group of moduli spaces of stable curves
AU - Amy Bradford
AU - Aitor Iribar López
AU - Andreas Kretschmer
AU - Hannah Larson
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14377
UR - https://arxiv.org/abs/2610.14377
AB - The new framework of semi-tautological extensions has allowed for explicit understanding of low-degree odd cohomology groups of moduli spaces of stable curves. Let S^*_ω(ℳ̅_g,n) ⊂ H^*(ℳ̅_g,n) denote the semi-tautological extension generated by H^11(ℳ̅_1,11), that is, the smallest system of subrings of H^*(ℳ̅_g,n) containing H^11(ℳ̅_1,11) and the tautological subrings, which is also closed under pullbacks along forgetful maps and pushforwards along gluing maps. Recently, it has been shown that H^11(ℳ̅_g,n) = S^11_ω(ℳ̅_g,n) and H^13(ℳ̅_g,n) = S^13_ω(ℳ̅_g,n) for all g and n. Meanwhile, H^15(ℳ̅_1,n) ≠ S_ω^15(ℳ̅_1,n) for all n ≥ 15. Nevertheless, we prove that H^15(ℳ̅_2,n) = S^15_ω(ℳ̅_2,n) for all n. Moreover, we determine all relations in S_ω^15(ℳ̅_g,n) for all g ≥ 2 and give explicit bases for this vector space in terms of decorated graphs. Conditional upon the conjectured dimension of H^15(ℳ̅_3,14), we also prove that H^15(ℳ̅_3,n) = S^15_ω(ℳ̅_3,n) for all n. As an application, we prove H^*(ℳ̅_g,n) = RH^*(ℳ̅_g,n) for several new small pairs (g,n).
ER -