How to Draw Tropical Planes

How to Draw Tropical Planes Sven Herrmann∗ Department of Mathematics Technische Universit¨at Darmstadt, Germany sherrmann@mathematik.tu-darmstadt.de Anders Jensen† Courant Research Center Georg-August-Universit¨at G¨ottingen, Germany jensen@uni-math.gwdg.de Michael Joswig‡ Department of Mathematics Technische Universit¨at Darmstadt, Germany joswig@mathematik.tu-darmstadt.de Bernd Sturmfels§ Department of Mathematics University of California, Berkeley, USA bernd@math.berkeley.edu Submitted: Sep 1, 2008; Accepted: Apr 14, 2009; Published: Apr 20, 2009 Mathematics Subject Classification: 52B40 (14M15, 05C05) Dedicated to Anders Bj¨orner on the occasion of his 60th birthday. Abstract The tropical Grassmannian parameterizes tropicalizations

of ordinary linear spaces, while the Dressian parameterizes all tropical linear spaces in TPn−1. We study these parameter spaces and we compute them explicitly for n ≤ 7. Planes are identified with matroid subdivisions and with arrangements of trees. These representations are then used to draw pictures. 1 Introduction A line in tropical projective space TPn−1 is an embedded metric tree which is balanced and has n unbounded edges pointing into the coordinate directions. The parameter space of these objects is the tropical Grassmannian Gr(2,n). This is a simplicial fan [29], known to evolutionary biologists as the space of phylogenetic trees with n labeled leaves [24, §3.5], and known to algebraic geometers as the moduli space of rational tropical curves [23]. ∗This author was supported by a Graduate Grant of TU Darmstadt. †This author was supported by a Sofia Kovalevskaja prize awarded to Olga Holtz at TU Berlin. ‡This author was supported by the DFG Research Unit “Polyhedral Surfaces”. §This author was supported by an Alexander-von-Humboldt senior award at TU Berlin and the US National Science Foundation. the electronic journal of combinatorics 16(2) (2009), #R6 1 Speyer [27, 28] introduced higher-dimensional tropical linear spaces. They are con- tractible polyhedral complexes...

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