Hostname: page-component-cb9f654ff-d5ftd Total loading time: 0 Render date: 2025-09-08T23:44:04.407Z Has data issue: false hasContentIssue false
  • English
  • Français

On Rational Equivalence in Tropical Geometry

Published online by Cambridge University Press:  20 November 2018

Lars Allermann ,
Simon Hampe  and
Johannes Rau
Lars Allermann
Affiliation:
Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany e-mail: allerman@mathematik.uni-kl.de
Simon Hampe
Affiliation:
Fachrichtung Mathematik, Universität der Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany e-mail: simon.hampe@math.uni-sb.de johannes.rau@math.uni-sb.de
Johannes Rau
Affiliation:
Fachrichtung Mathematik, Universität der Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany e-mail: simon.hampe@math.uni-sb.de johannes.rau@math.uni-sb.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This article discusses the concept of rational equivalence in tropical geometry (and replaces an older, 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 ${{\mathbf{R}}^{n}}$ 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 ${{\mathbf{R}}^{n}}$ 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

14T05tropical geometryrational equivalence

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 2 , 01 April 2016 , pp. 241 - 257
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Allermann, L. and Rau, J., Tropical rational equivalence on ℝr. http://arxiv:0811.2860 Google Scholar
[2] Allermann, L. and Rau, J., First steps in tropical intersection theory. Math. Z. 264(2010), no. 3, 633–670.http://dx.doi.org/10.1007/s00209-009-0483-1 Google Scholar
[3] Baker, M. and Norine, S., Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215(2007), no. 2, 766–788.http://dx.doi.Org/10.1016/j.aim.2007.04.012 Google Scholar
[4] Cools, F., Draisma, J., Payne, S., and Robeva, E., A tropical proof of the Brill-Noether theorem. Adv. Math. 230(2012), no. 2, 759–776.http://dx.doi.Org/10.1016/j.aim.2O12.02.019 Google Scholar
[5] Fulton, W., Introduction to toric varieties. The 1989 William H. Roever lectures in geometry. Princeton University Press, Princeton, NJ, 1993.Google Scholar
[6] Fulton, W. and Sturmfels, B., Intersection theory on toric varieties. Topolofy 36(1997), no. 2,335–353.http://dx.doi.Org/10.1016/0040-9383(96)00016-X Google Scholar
[7] Gathmann, A. and Kerber, M., A Riemann-Roch theorem in tropical geometry. Math. Z. 259(2008), no. 1, 217–230. http://dx.doi.org/10.1007/s00209-007-0222-4 Google Scholar
[8] Gathmann, A., Kerber, M., and Markwig, H., Tropical fans and the moduli spaces of tropical curves. Compos. Math. 145(2009), no. 1, 173–195.http://dx.doi.Org/10.1112/S0010437X08003837 Google Scholar
[9] Haase, C., Musiker, G., Yu, J., Linear systems on tropical curves. Math. Z. 270(2012), no. 3–4, 1111–1140. http://dx.doi.org/10.1007/s00209-011-0844-4 Google Scholar
[10] Jensen, A. and Yu, J., Stable intersections of tropical varieties. arxiv:1309.7064 Google Scholar
[11] Katz, E., Tropical intersection theory from toric varieties. Collect. Math. 63(2012), no. 1, 29–44.http://dx.doi.Org/10.1007/s13348-010-0014-8 Google Scholar
[12] Markwig, H. and Johannes, R., Tropical descendant Gromov-Witten invariants. Manuscr. Math. 129(2009), no. 3, 293–335. http://dx.doi.org/10.1007/s00229-009-0256-5 Google Scholar
[13] McMullen, P., The polytope algebra. Adv. Math. 78(1989), no. 1, 76–130.http://dx.doi.Org/10.1016/0001-8708(89)90029-7 Google Scholar
[14] Mikhalkin, G., Tropical geometry and its applications. In: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures), European Mathematical Society (EMS), Zürich, 2006, pp. 827–852.Google Scholar
[15] Mikhalkin, G. and Rau, J., Tropical geometry. ICM publication, in preparation, http://www.math.uni-sb.de/ag-rau/-news Google Scholar
[16] Mikhalkin, G. and Zharkov, I., Tropical curves, their Jacobians and theta functions. In: Curves and abelian varieties, Proceedings of the international conference, Athens, GA, USA, March 30-April 2, 2007, American Mathematical Society, Providence, RI, 2008, pp. 203–230.Google Scholar
[17] Rau, J., Intersections on tropical moduli spaces. Rocky Mountain J. Math., to appear. arxiv:0812.3678 Google Scholar
[18] Rau, J., Tropical intersection theory and gravitational descendants.PhD Thesis, Technische Universität Kaiserslautern, 2009.http://kluedo.ub.uni-kl.de/volltexte/2009/2370/ Google Scholar
[19] Richter-Gebert, J., Sturmfels, B., and Theobald, T., First steps in tropical geometry. In: Idempotent mathematics and mathematical physics, Proceedings of the international workshop, Vienna, Austria, February 3-10, 2003, American Mathematical Society, Providence, RI, 2005, pp. 289–317.Google Scholar