Bier Spheres and Toric Topology

We compute the real and complex Buchstaber numbers of an arbitrary Bier sphere. In dimension two, we identify all the 13 different combinatorial types of Bier spheres and show that 12 of them are nerve complexes of nestohedra, while the remaining one is a nerve complex of a generalized permutohedron. As an application of our results, we construct a regular normal fan for each of those 13 Delzant polytopes, compute the cohomology rings of the corresponding nonsingular projective toric varieties, and examine the orientability of the corresponding small covers.