Toric varieties constructed from atomic lattices

Every finite lattice L determines certain commutative graded algebras D parametrized by some special subsets G of L (so called building sets). These algebras are generalizations of the cohomology algebras of hyperplane arrangement compactifications that appeared in work of De Concini and Procesi. We find a Gr\"obner basis of the relation ideal of D. The main result of the talk is an interpretation of D (in the case where L is atomic) as the Chow ring of a smooth toric variety constructed from L and G. We describe this variety both by its fan and geometrically, by a series of blowups and orbit removal. All the new results of the talk have been obtained jointly with Eva Feichtner.