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Combinatorial Aspects of Commutative Algebra and Algebraic Geometry pp 73–88Cite as

Realization Spaces for Tropical Fans

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Part of the book series: Abel Symposia ((ABEL,volume 6))

Abstract

We introduce a moduli functor for varieties whose tropicalization realizes a given weighted fan and show that this functor is an algebraic space in general, and is represented by a scheme when the associated toric variety is quasiprojective. We study the geometry of these tropical realization spaces for the matroid fans studied by Ardila and Klivans, and show that the tropical realization space of a matroid fan is a torus torsor over the realization space of the matroid. As a consequence, we deduce that these tropical realization spaces satisfy Murphy’s Law.

Sam Payne is supported by the Clay Mathematics Institute.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA

    Eric Katz

  2. Mathematics, Stanford University, Bldg. 380, 450 Serra Mall, Stanford, CA, 94305, USA

    Sam Payne

Authors
  1. Eric Katz
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  2. Sam Payne
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Correspondence to Eric Katz .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen, 5008, Norway

    Gunnar Fløystad

  2. Dept. Mathematics & Statistics, University of Tromsø, Tromsø, 9037, Norway

    Trygve Johnsen

  3. Dept. Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen, 5008, Norway

    Andreas Leopold Knutsen

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Katz, E., Payne, S. (2011). Realization Spaces for Tropical Fans. In: Fløystad, G., Johnsen, T., Knutsen, A. (eds) Combinatorial Aspects of Commutative Algebra and Algebraic Geometry. Abel Symposia, vol 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19492-4_6

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