Abstract
Let E be a plane in an algebraic torus \( (K^*)^n \) over an algebraically closed field K. Given a balanced 1-dimensional fan C in the tropicalization of E, i. e. in the Bergman fan of the corresponding matroid, we give a complete algorithmic answer to the question whether or not C can be realized as the tropicalization of an algebraic curve contained in E. Moreover, in the case of realizability the algorithm also determines the dimension of the moduli space of all algebraic curves in E tropicalizing to C, a concrete simple example of such a curve, and whether C can also be realized by an irreducible algebraic curve in E. In the first important case when E is a general plane in a 3-dimensional torus we also use our algorithm to prove some general criteria for C that imply its realizability resp. non-realizability. They include and generalize the main known obstructions by Brugallé-Shaw and Bogart-Katz coming from tropical intersection theory.
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)
Ardila, F., Klivans, C.: The Bergman complex of a matroid and phylogenetic trees. J. Comb. Theory Ser. B 96(1), 38–49 (2006)
Bieri, R., Groves, J.: A rigidity property for the set of all characters induced by valuations. Trans. Am. Math. Soc. 294, 425–434 (1986)
Birkmeyer, A.L.: realizationMatroids.lib—a Singular library for relative realizability questions on tropical curves. http://www.mathematik.uni-kl.de/~gathmann/matroids.php (2012)
Birkmeyer, A.L., Gathmann, A.: Realizability of tropical curves in a plane in the non-constant coefficient case. http://arxiv.org/abs/1412.3035 (2014)
Bogart, T., Katz, E.: Obstructions to lifting tropical curves in surfaces in 3-space. SIAM J. Discrete Math. 26, 1050–1067 (2012)
Brodsky, S., Joswig, M., Morrison, R., Sturmfels, B.: Moduli of tropical plane curves. Res. Math. Sci. 2 (2015)
Brugallé, E., Shaw, K.: Obstructions to approximating tropical curves in surfaces via intersection theory. Can. J. Math. 67(3), 527–572 (2015)
Cartwright, D., Dudzik, A., Manjunath, M., Yao, Y.: Embeddings and immersions of tropical curves. Collect. Math. 67(1), 1–19 (2016)
Cox, D., Little, J., Schenck, H.: Toric Varieties. American Mathematical Society (AMS), Providence, RI (2011)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular—a computer algebra system for polynomial computations. http://www.singular.uni-kl.de
Einsiedler, M., Kapranov, M., Lind, D.: Non-Archimedean amoebas and tropical varieties. J. Reine Angew. Math. 601, 139–157 (2006)
Eisenbud, D.: Commutative Algebra with a View Towards Algebraic Geometry. Springer, New York (1995)
François, G.: Tropical intersection products and families of tropical curves. Ph.D. thesis, University of Kaiserslautern (2012). https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3350
François, G., Rau, J.: The diagonal of tropical matroid varieties and cycle intersections. Collect. Math. 64(2), 185–210 (2013)
Fulton, W.: Intersection Theory. Springer, Berlin (1998)
Fulton, W., Sturmfels, B.: Intersection theory on toric varieties. Topology 36(2), 335–353 (1997)
Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli space of rational tropical curves. Compos. Math. 145(1), 173–195 (2009)
Hartshorne, R.: Algebraic Geometry. Springer, Berlin (1977)
Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. American Mathematical Society (AMS), Providence, RI (2015)
Mikhalkin, G.: Decomposition into pairs-of-pants for complex algebraic hypersurfaces. Topology 43(5), 1035–1065 (2004)
Nishinou, T., Siebert, B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135, 1–51 (2006)
Oxley, J.: Matroid Theory. Oxford University Press, Oxford (1992)
Rau, J.: Intersections on tropical moduli spaces. http://arxiv.org/abs/0812.3678 (2008)
Rau, J.: Tropical intersection theory and gravitational descendants. Ph.D. thesis, University of Kaiserslautern (2009). https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2122
Shaw, K.: A tropical intersection product in matroidal fans. SIAM J. Discrete Math. 27(1), 459–491 (2013)
Speyer, D.: Tropical geometry. Ph.D. thesis, University of California, Berkeley (2005). http://www-personal.umich.edu/~speyer/thesis.pdf
Speyer, D.: Uniformizing tropical curves I: genus zero and one. Algebra Number Theory 8, 963–998 (2014)
Sturmfels, B.: Solving systems of polynomial equations. In: CBMS Regional Conference Series in Mathematics 97. American Mathematical Society, New York (2002)
Sturmfels, B., Tevelev, J.: Elimination theory for tropical varieties. Math. Res. Lett. 15(3), 543–562 (2008)
Ziegler, G.M.: Lectures on Polytopes. Springer-Verlag, New York (2002)
Acknowledgments
Kirsten Schmitz has been supported by the DFG Grant Ga 636/3.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: János Pach
Rights and permissions
About this article
Cite this article
Birkmeyer, A.L., Gathmann, A. & Schmitz, K. The Realizability of Curves in a Tropical Plane. Discrete Comput Geom 57, 12–55 (2017). https://doi.org/10.1007/s00454-016-9816-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-016-9816-0