%0 Generic %1 FeichtnerSturmfels2004 %A Feichtner, Eva Maria %A Sturmfels, Bernd %D 2004 %K Ardila Bergman De_Concini Feichtner Klivans Procesi Sturmfels arrangement compactification complex fan hyperplane matroid order polytope simplex tropics variety %T Matroid polytopes, nested sets and Bergman fans %U http://arxiv.org/abs/math/0411260 %X The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila-Klivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De Concini-Procesi compactification of the complement of a hyperplane arrangement, while the Bergman fan specifies the tropical compactification. These two compactifications are almost equal, and we highlight the subtle differences.

@misc{FeichtnerSturmfels2004, abstract = { The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila-Klivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De Concini-Procesi compactification of the complement of a hyperplane arrangement, while the Bergman fan specifies the tropical compactification. These two compactifications are almost equal, and we highlight the subtle differences. }, added-at = {2008-12-23T16:12:05.000+0100}, author = {Feichtner, Eva Maria and Sturmfels, Bernd}, biburl = {https://www.bibsonomy.org/bibtex/21001bbafa9b7dd5641c0ad68d9d8f2ed/fwd13}, description = {Matroid polytopes, nested sets and Bergman fans}, interhash = {47a1d3a5dce127cce53db13d3406b9ba}, intrahash = {1001bbafa9b7dd5641c0ad68d9d8f2ed}, keywords = {Ardila Bergman De_Concini Feichtner Klivans Procesi Sturmfels arrangement compactification complex fan hyperplane matroid order polytope simplex tropics variety}, note = {cite arxiv:math/0411260 Comment: 22 pages, 3 figures}, timestamp = {2008-12-23T17:30:36.000+0100}, title = {Matroid polytopes, nested sets and Bergman fans}, url = {http://arxiv.org/abs/math/0411260}, year = 2004 }