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Combinatorics of tropical Hurwitz cycles. (English) Zbl 1335.14015
Author’s abstract: We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri, and Markwig [A. Bertram, J. Comb. Theory, Ser. A 120, No. 7, 1604–1631 (2013; Zbl 1317.14136)]. We show that all such loci are connected in codimension one. If we mark presages of simple ramification points, then for a generic choice of such points, the resulting cycles are weakly irreducible, i.e. an integer multiple of an irreducible cycle. We study how Hurwitz cycles can be written as divisors of rational functions and show that they are numerically equivalent to a tropical version of a representation as a some of boundary divisors. The results and counterexamples in this paper were obtained with the help of a-tint, an extension for polymake for tropical intersection theory.
Reviewer: Thanh Vu (Lincoln)

MSC:
14T05 Tropical geometry (MSC2010)
Software:
a-tint; polymake
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References:
[1] Allermann, L., Hampe, S., Rau, J.: On rational equivalence in tropical geometry (2014). arXiv:1408.1537 (preprint) · Zbl 1407.14057
[2] Ardila, F., Klivans, C.: The Bergman complex of a matroid and phylogenetic trees. J. Comb. Theory Ser. B. 96, 38-49 (2006). arXiv:math/0311370v2 · Zbl 1082.05021
[3] Bertrand, B., Brugallée, E., Mikhalkin, G.: Tropical open Hurwitz numbers. Rend. Semin. Mat. Univ. Padova. 125, 157-171 (2011). arXiv:1005.4628 · Zbl 1226.14066
[4] Bertram, A; Cavalieri, R; Markwig, H, Polynomiality, wall crossings and tropical geometry of rational double Hurwitz cycles, JCTA, 120, 1604-1631, (2013) · Zbl 1317.14136
[5] Berstein, I; Edmonds, AL, On the classification of generic branched coverings of surfaces, Ill. J. Math., 28, 64-82, (1984) · Zbl 0551.57001
[6] Bogart, T; Jensen, AN; Speyer, D; Sturmfels, B; Thomas, RR, Computing tropical varieties, J. Symb. Comput., 42, 54-73, (2007) · Zbl 1121.14051
[7] Clebsch, A, Zur theorie der riemann’schen fläche, Math. Ann., 6, 216-230, (1873) · JFM 05.0227.01
[8] Cavalieri, R., Johnson, P., Markwig, H.: Tropical Hurwitz numbers, J. Algebr. Comb. 32(2), 241-265 (2010). arXiv:0804.0579 · Zbl 1218.14058
[9] Cavalieri, R., Markwig, H., Ranganathan, D.: Tropical compactification and the gromov-witten theory of \(\mathbb{P}^1\) (2014). arXiv:1410.2837 (preprint) · Zbl 1391.14111
[10] Cartwright, D., Payne, S.: Connectivity of tropicalizations. Math. Res. Lett. 19(5), 1089-1095 (2012). arXiv:1204.6589 · Zbl 1291.14091
[11] Ekedahl, T; Lando, S; Shapiro, M; Vainshtein, A, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math., 146, 297-327, (2001) · Zbl 1073.14041
[12] Francois, G, Cocycles on tropical varieties via piecewise polynomials, Proc. Am. Math. Soc., 141, 481-497, (2013) · Zbl 1295.14058
[13] Francois, G; Rau, J, The diagonal of tropical matroid varieties and cycle intersections, Collect. Math., 64, 185-210, (2013) · Zbl 1312.14144
[14] Feichtner, EM; Sturmfels, B, Matroid polytopes, nested sets and Bergman fans, Port. Math. Nova Série, 62, 437-468, (2005) · Zbl 1092.52006
[15] Fulton, W; Sturmfels, B, Intersection theory on toric varieties, Topology, 36, 335-353, (1997) · Zbl 0885.14025
[16] Graber, T., Harris, J., Starr, J.: A note on Hurwitz schemes of covers of a positive genus curve (2002). arXiv:math/0205056 (preprint) · Zbl 1295.14058
[17] Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes. In: Polytopes: Combinatorics and Computation (Oberwolfach, 1997), pp. 43-73. (2000) · Zbl 0960.68182
[18] Goulden, I., Jackson, D., Vakil, R.: Towards the geometry of double Hurwitz numbers. Adv. Math. 198(1), 43-92 (2005). arXiv:math/0309440 · Zbl 1086.14022
[19] Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli spaces of tropical curves. Compos. Math. 145, 173-195 (2009). arXiv:0708.2268 · Zbl 1169.51021
[20] Gibney, A., Maclagan, D.: Equations for Chow and Hilbert quotients. Algebra Number Theory. 7, 855-885 (2010). arXiv:0707.1801 · Zbl 1210.14051
[21] Graber, T., Vakil, R.: Relative virtual localization and vanishing of tautological classes on moduli spaces of curves. Duke Math. J. 130(1), 1-37 (2005). arXiv:math/0309227 · Zbl 1088.14007
[22] Hampe, S.: A-tint: a polymake extension for algorithmic tropical intersection theory. Eur. J. Comb. 36, 579-607 (2014). arXiv:1208.4248 · Zbl 1285.14071
[23] Hurwitz, A, Über riemann’sche flächen mit gegebenen verzweigungspunkten, Math. Ann., 39, 1-61, (1891) · JFM 23.0429.01
[24] Johnson, P.: Hurwitz numbers, ribbon graphs and tropicalizations. Contemp. Math. 580, 55-72 (2012). arXiv:1303.1543 · Zbl 1317.14118
[25] Kanev, V, Irreducible components of Hurwitz spaces parameterizing Galois coverings of curves of positive genus, Pure Appl. Math. Q., 10, 193-222, (2014) · Zbl 1338.14031
[26] Kontsevich, M, Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys., 147, 1-23, (1992) · Zbl 0756.35081
[27] Kerber, M., Markwig, H.: Intersecting psi-classes on tropical \(\cal M_{0,n}\). Int. Math. Res. Not. (2008). arXiv:0709.3953 · Zbl 1205.14070
[28] Mikhalkin, G.: Enumerative tropical algebraic geometry in \(\mathbb{R}^2\). J. Am. Math. Soc. 18(2), 313-377 (2005). arXiv:math/0312530 · Zbl 1092.14068
[29] Mikhalkin, G.: Moduli spaces of rational tropical curves. In: Proceedings of Gökova Geometry: Topology Conference 2006, pp. 39-51. (2007) · Zbl 1203.14027
[30] Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence, RI (2015) · Zbl 1321.14048
[31] Nishinou, T., Siebert, B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135(1), 1-51 (2006). arXiv:math/0409060 · Zbl 1105.14073
[32] Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz numbers and matrix models. In: I (2001). arXiv:math/0101147 · Zbl 1205.14072
[33] Rau, J.: Tropical intersection theory and gravitational descendants. Ph.D. Thesis (2009)
[34] Severi, F.: Vorlesungen über algebraische Geometrie. Teubner-Verlag, Berlin (1921) · JFM 48.0687.01
[35] Speyer, D.: Tropical linear spaces. SIAM J. Discrete Math. 22, 1527-1558 (2008). arXiv:math/0410455 · Zbl 1191.14076
[36] Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4(3), 389-411 (2004). arXiv:math/0304218 · Zbl 1065.14071
[37] Vetro, F, Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type \(D_d\), Manuscr. Math., 125, 353-368, (2008) · Zbl 1139.14023
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