zbMATH — the first resource for mathematics

Lifting tropical self intersections. (English) Zbl 1441.14205
Tropical geometry is a combinatorially effective tool, which can be viewed as a piecewise-linear shadow classical algebraic geometry. In the article under review, the tropicalizations of intersections of plane curves are studied, focusing on the case where two plane curves have the same tropicalization. For a comprehensive study of tropical divisors see [M. Baker and D. Jensen, in: Nonarchimedean and tropical geometry. Based on two Simons symposia, Island of St. John, March 31 – April 6, 2013 and Puerto Rico, February 1–7, 2015. Cham: Springer. 365–433 (2016; Zbl 1349.14193)].
The main result of the article shows that tropical divisors arising as such intersections form a polyhedral complex, and the author computes the dimension of this complex. Specializing to the case when the genus is at most \(1\), it is also shown that a large class of divisors satisfying certain combinatorial criteria are realisable.

14T15 Combinatorial aspects of tropical varieties
Full Text: DOI arXiv
[1] Allermann, Lars; Rau, Johannes, First steps in tropical intersection theory, Math. Z., 264, 3, 633-670 (2010) · Zbl 1193.14074
[2] Baker, M.; Jensen, D., Degeneration of Linear Series from the Tropical Point of View and Applications, 365-433 (2016), Springer International Publishing: Springer International Publishing Cham · Zbl 1349.14193
[3] Brugallé, E.; Medrano, L., Inflection points of real and tropical plane curves, J. Singul., 4, 74-103 (2011) · Zbl 1292.14042
[4] Brugallé, E.; Itenberg, I.; Mikhalkin, G.; Shaw, K., Brief introduction to tropical geometry, (Proceedings of the Gökova Geometry-Topology Conference 2014. Proceedings of the Gökova Geometry-Topology Conference 2014, Gökova Geometry/Topology Conference (GGT), Gökova (2015)), 1-75 · Zbl 1354.14089
[5] Burgos Gil, J.; Sombra, M., When do the recession cones of a polyhedral complex form a fan?, Discrete Comput. Geom., 46, 4, 789-798 (2011) · Zbl 1233.14031
[6] Caporaso, L.; Len, Y.; Melo, M., Algebraic and combinatorial rank of divisors on finite graphs, J. Math. Pures Appl. (9), 104, 2, 227-257 (2015) · Zbl 1402.14033
[7] Cartwright, D.; Jensen, D.; Payne, S., Lifting divisors on a generic chain of loops, Canad. Math. Bull., 58, 2, 250-262 (2015) · Zbl 1327.14266
[8] Cavalieri, R.; Markwig, H.; Ranganathan, D., Tropicalizing the space of admissible covers, Math. Ann., 364, 3-4, 1275-1313 (2016) · Zbl 1373.14064
[9] Cueto, M. A.; Markwig, H., How to repair tropicalizations of plane curves using modifications, Exp. Math., 25, 2, 130-164 (2016) · Zbl 1349.14196
[10] Eisenbud, D.; Harris, J., 3264 and All That: A Second Course in Algebraic Geometry (2016), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1341.14001
[11] Haase, C.; Musiker, G.; Yu, J., Linear systems on tropical curves, Math. Z., 270, 3-4, 1111-1140 (2012) · Zbl 1408.14201
[12] He, X., A generalization of lifting non-proper tropical intersections (2016), preprint
[13] Ilten, N.; Len, Y., Projective duals to algebraic and tropical hypersurfaces, Proc. Lond. Math. Soc., 119, 1234-1278 (2019)
[14] Len, Y.; Markwig, H., Lifting tropical bitangents, J. Symbolic Comput., 96, 122-152 (2019) · Zbl 07074721
[15] Len, Y.; Ranganathan, D., Enumerative geometry of elliptic curves on toric surfaces, Israel J. Math., 226, 351-385 (2018) · Zbl 1402.14081
[16] Maclagan, D.; Sturmfels, B., Introduction to Tropical Geometry (2015), American Mathematical Society (AMS): American Mathematical Society (AMS) Providence, RI · Zbl 1321.14048
[17] Markwig, H., The Enumeration of Plane Tropical Curves (2006), Technische Universität Kaiserslautern, PhD thesis
[18] Mikhalkin, G.; Zharkov, I., Tropical curves, their Jacobians and theta functions, (Curves and Abelian Varieties. Curves and Abelian Varieties, Contemp. Math., vol. 465 (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 1152.14028
[19] Mikhalkin, G., Tropical geometry and its applications, (Sanz-Sole, M.; etal., Invited Lectures v. II, Proceedings of the ICM Madrid (2006)), 827-852 · Zbl 1103.14034
[20] Moeller, M.; Ulirsch, M.; Werner, A., Realizability of tropical canonical divisors (2017), preprint
[21] Morrison, R., Tropical images of intersection points, Collect. Math., 66, 2, 273-283 (2015) · Zbl 1331.14059
[22] Nicaise, J.; Payne, S.; Schroeter, F., Tropical refined curve counting via motivic integration (2016), preprint · Zbl 1430.14037
[23] Nisse, M.; Sottile, F., Higher convexity for complements of tropical varieties, Math. Ann., 365, 1, 1-12 (2016) · Zbl 1369.14072
[24] Osserman, B.; Payne, S., Lifting tropical intersections, Doc. Math., 18, 121-175 (2013) · Zbl 1308.14069
[25] Osserman, B.; Rabinoff, J., Lifting nonproper tropical intersections, (Tropical and Non-Archimedean Geometry. Tropical and Non-Archimedean Geometry, Contemp. Math., vol. 605 (2013), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 15-44 · Zbl 1320.14078
[26] Rau, J., Intersections on tropical moduli spaces, Rocky Mountain J. Math., 46 (2009)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.