Abstract
We define an intersection product of tropical cycles on matroid varieties (via cutting out the diagonal) and show that it is well-behaved. In particular, this enables us to intersect cycles on moduli spaces of tropical rational marked curves \(\mathcal M _n\) and \(\mathcal M _n^{\text{ lab}}(\Delta , \mathbb R ^r)\). This intersection product can be extended to smooth varieties (whose local models are matroid varieties). We also study pull-backs of cycles and rational equivalence.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allermann, L., Rau, J.: First steps in tropical intersection theory. Math. Z. 264(3), 633–670 (2010). arxiv:0709.3705v3
Allermann, L., Rau, J.: Tropical rational equivalence on \(R^r\). arxiv:0811.2860v2
Allermann, L.: Tropical intersection products on smooth varieties. J. Eur. Math. Soc. 14(1), 107–126 (2012). arxiv:0904.2693v2
Allermann, L.: Tropical intersection theory. Ph.D. thesis, Technische Universität Kaiserslautern (2010). http://kluedo.ub.uni-kl.de/volltexte/2010/2469
Ardila, F., Klivans, C.J.: The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory B 96(1), 38–49 (2006). arxiv:math/0311370v2
Feichtner, E.M., Sturmfels, B.: Matroid polytopes, nested sets and Bergman fans. Port. Math. (N.S.) 62(4), 437–468 (2005). arxiv:math/0411260
Fink, A.: Tropical cycles and Chow polytopes. arxiv:1001.4784
Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli spaces of tropical curves. Compos. Math. 145(1), 173–195 (2009). arxiv:0708.2268
Mikhalkin, G.: Tropical geometry and its applications.In: International Congress of Mathematicians, vol. II, pp. 827–852. European Mathematical Society, Zurich (2006). arxiv:math/0601041
Oxley, J.G.: Matroid Theory. Oxford University Press, Oxford (1992)
Rau, J.: Tropical intersection theory and gravitational descendants. Ph.D. thesis, Technische Universitat Kaiserslautern (2009). http://kluedo.ub.uni-kl.de/volltexte/2009/2370
Shaw, K.M.: A tropical intersection product in matroidal fans. arxiv:1010.3967
Speyer, D.E.: Tropical linear spaces. SIAM J. Discrete Math. 22(4), 1527–1558 (2008). arxiv: math/0410455
Sturmfels, B.: Solving systems of polynomial equations. In: CBMS Regional Conference Series in Mathematics, vol. 97. Conference Board of the Mathematical Sciences, Washington, DC (2002)
White, N.: Theory of Matroids. Encyclopedia Math. Appl., vol. 26. Cambridge University Press, London (1986)
Acknowledgments
We are grateful to Kristin Shaw, Federico Ardila and Andreas Gathmann for many helpful discussions and comments. We would also like to thank the reviewer for useful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
G. François is supported by the Fonds national de la Recherche (FNR), Luxembourg.
Rights and permissions
About this article
Cite this article
François, G., Rau, J. The diagonal of tropical matroid varieties and cycle intersections. Collect. Math. 64, 185–210 (2013). https://doi.org/10.1007/s13348-012-0072-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-012-0072-1