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On Rational Equivalence in Tropical Geometry

  Published:2015-12-24
 Printed: Apr 2016
  • Lars Allermann,
    Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
  • Simon Hampe,
    Fachrichtung Mathematik, Universität der Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany
  • Johannes Rau,
    Fachrichtung Mathematik, Universität der Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany
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Abstract

This article discusses the concept of rational equivalence in tropical geometry (and replaces an older and imperfect version). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the ``bounded'' Chow groups of $\mathbb{R}^n$ by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest: We show that every tropical cycle in $\mathbb{R}^n$ is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).
Keywords: tropical geometry, rational equivalence tropical geometry, rational equivalence
MSC Classifications: 14T05 show english descriptions Tropical geometry [See also 12K10, 14M25, 14N10, 52B20] 14T05 - Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]
 

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