PORTUGALIAE MATHEMATICA Vol. 62, No. 4, pp. 437468 (2005) 

Matroid polytopes, nested sets and Bergman fansEva Maria Feichtner and Bernd SturmfelsDepartment of Mathematics, ETH Zürich,8092 Zürich  SWITZERLAND Email: feichtne@math.ethz.ch Department of Mathematics, UC Berkeley, Berkeley CA 94720  USA Email: bernd@math.berkeley.edu Abstract: The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a selfcontained 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 ArdilaKlivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De ConciniProcesi 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. Classification (MSC2000): 52B20.; 05B35, 14D99, 52B40, 52C35. Full text of the article:
Electronic version published on: 7 Mar 2008.
© 2005 Sociedade Portuguesa de Matemática
