Cellular 𝔸^1-homology of wonderful models of subspace arrangements

Abstract

We compute the cellular 𝔸¹-homology of De Concini--Procesi wonderful models of subspace arrangements. For a building set 𝒢 over a field k, we identify the cellular 𝔸¹-chain complex of ℙ(𝒢) with an η-twisted nested-set complex carrying Milnor--Witt coefficients and derived orientation data. The key geometric input is a motivic blow-up calculation: for a blow-up along a smooth center of codimension c, the relevant connecting class is (c−1)_εη, hence it is zero for c odd and η for c even. This replaces the parity condition in the computation of Rains by a Milnor--Witt attaching class. As a consequence, the part of cellular 𝔸¹-homology surviving after multiplication by η, and also the homology after inverting η, are expressed by the interval cohomology of the $2$-divisible subposet of the lattice generated by 𝒢. For the braid arrangement, the condition becomes the odd-block condition on partitions, yielding explicit decompositions for the cellular 𝔸¹-homology of ℳ̅_0,N and examples in low rank.

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BibTeX RIS

BibTeX

@misc{liu2026cellular,
  title = {{Cellular $\mathbbA^1$-homology of wonderful models of subspace arrangements}},
  author = {Haoyang Liu and Keyao Peng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.29097},
  eprint = {2607.29097},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.29097},
}

RIS

TY  - UNPB
TI  - Cellular 𝔸^1-homology of wonderful models of subspace arrangements
AU  - Haoyang Liu
AU  - Keyao Peng
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2607.29097
UR  - https://arxiv.org/abs/2607.29097
AB  - We compute the cellular 𝔸^1-homology of De Concini--Procesi wonderful models of subspace arrangements. For a building set 𝒢 over a field k, we identify the cellular 𝔸^1-chain complex of ℙ(𝒢) with an η-twisted nested-set complex carrying Milnor--Witt coefficients and derived orientation data. The key geometric input is a motivic blow-up calculation: for a blow-up along a smooth center of codimension c, the relevant connecting class is (c−1)_εη, hence it is zero for c odd and η for c even. This replaces the parity condition in the computation of Rains by a Milnor--Witt attaching class. As a consequence, the part of cellular 𝔸^1-homology surviving after multiplication by η, and also the homology after inverting η, are expressed by the interval cohomology of the $2$-divisible subposet of the lattice generated by 𝒢. For the braid arrangement, the condition becomes the odd-block condition on partitions, yielding explicit decompositions for the cellular 𝔸^1-homology of ℳ̅_0,N and examples in low rank.
ER  -

Cellular 𝔸¹-homology of wonderful models of subspace arrangements

Abstract

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