## The homology of real subspace arrangements.(English)Zbl 1213.14102

Let $$V$$ be a vector space and let $$G$$ be a building set, i.e. a finite collection of subspaces of the dual $$V^*$$, whose elements are indecomposable. The open set $$V-\bigcup_{H_i\in G} H_i^\perp$$ has a natural emebdding in the product of the projective spaces $$\mathbb P(V/H_i^\perp)$$. The closure $$Y_G$$ of the image is the De Concini–Procesi model of $$G$$. The variety $$Y_G$$, in the case where $$G$$ is a braid arrangement, is connected with the real part of the closure of the moduli space $$\bar M_{0,n}(\mathbb R)$$ of marked rational curves. Starting with a combinatorial description of the homology of $$V-\bigcup_{H_i\in G} H_i^\perp$$, it is possible to characterize the homology of $$\bar M_{0,n}(\mathbb R)$$. The author performs a similar analysis when $$G$$ is a general building set. Using chains of blow down of real De Concini - Procesi models, the author obtains a description of the ring structure of the homology of $$Y_G$$. By using this method, the author also proves that the homology of $$\bar M_{0,n}(\mathbb R)$$ has no odd torsion.

### MSC:

 14N20 Configurations and arrangements of linear subspaces 14F25 Classical real and complex (co)homology in algebraic geometry
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