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PreprintarXiv licencearXiv

The integral Chow rings of 𝒜3 and ℳ3

  • Emma Bishop
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Abstract

Over ℂ, we compute the integral Chow ring of 𝒜3, the moduli stack of principally polarized abelian threefolds. Along the way, we compute the integral Chow ring of ℳ3, the moduli stack of smooth genus-three curves. The ring CH(𝒜3) is generated by the Chern classes of the Hodge bundle, along with three classes supported on the locus of decomposable abelian varieties, subject to 16 relations. The ring CH(ℳ3) is generated by the Chern classes of the Hodge bundle, with five relations. In particular, we find that the top Chern class of the Hodge bundle, λ3, has order 504 on 𝒜3 and 252 on ℳ3. For 𝒜3, this removes the factor of 4 from the bound 252|ord(λ3)| 1008 of Ekedahl and van der Geer.

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Cite

BibTeXRIS
BibTeX
@misc{bishop2026integral,
  title = {{The integral Chow rings of $\mathcal A_3$ and $\mathcal M_3$}},
  author = {Emma Bishop},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06014},
  eprint = {2611.06014},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06014},
}
RIS
TY  - UNPB
TI  - The integral Chow rings of 𝒜_3 and ℳ_3
AU  - Emma Bishop
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06014
UR  - https://arxiv.org/abs/2611.06014
AB  - Over ℂ, we compute the integral Chow ring of 𝒜_3, the moduli stack of principally polarized abelian threefolds. Along the way, we compute the integral Chow ring of ℳ_3, the moduli stack of smooth genus-three curves. The ring CH(𝒜_3) is generated by the Chern classes of the Hodge bundle, along with three classes supported on the locus of decomposable abelian varieties, subject to 16 relations. The ring CH(ℳ_3) is generated by the Chern classes of the Hodge bundle, with five relations. In particular, we find that the top Chern class of the Hodge bundle, λ_3, has order 504 on 𝒜_3 and 252 on ℳ_3. For 𝒜_3, this removes the factor of 4 from the bound 252|ord(λ_3)| 1008 of Ekedahl and van der Geer.
ER  -