2026/07/31 by Haoyang Liu, Keyao Peng
Mathematics · #math.AG #math.AT
arxiv created 2026/07/31 · arxiv updated 2026/08/03
We compute the cellular \mathbbA1-homology of De Concini--Procesi wonderful models of subspace arrangements. For a building set G over a field k, we identify the cellular \mathbbA1-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 \mathbbA1-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 \mathbbA1-homology of \mathcal M0,N and examples in low rank.