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

Rains, Eric M.

J. Topol. 3, No. 4, 786-818 (2010).

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.

Reviewer: Luca Chiantini (Siena)

MSC:

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