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Tropical moduli spaces of stable maps to a curve. (English) Zbl 1400.14146
Böckle, Gebhard (ed.) et al., Algorithmic and experimental methods in algebra, geometry, and number theory. Cham: Springer (ISBN 978-3-319-70565-1/hbk; 978-3-319-70566-8/ebook). 287-309 (2017).
Summary: We construct moduli spaces of rational covers of an arbitrary smooth tropical curve in \({\mathbb R}^r\) as tropical varieties. They are contained in the balanced fan parametrizing tropical stable maps of the appropriate degree to \({\mathbb R}^r\). The weights of the top-dimensional polyhedra are given in terms of certain lattice indices and local Hurwitz numbers.
For the entire collection see [Zbl 1394.14002].

14T05 Tropical geometry (MSC2010)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
51M20 Polyhedra and polytopes; regular figures, division of spaces
a-tint; GAP; polymake
Full Text: DOI arXiv
[1] L. Allermann, Tropical intersection products on smooth varieties. J. Eur. Math. Soc. 14(1), 107-126 (2012) · Zbl 1256.14069
[2] L. Allermann, J. Rau, First steps in tropical intersection theory. Math. Z. 264(3), 633-670 (2010). arXiv:0709.3705 · Zbl 1193.14074
[3] K. Behrend, Gromov-Witten invariants in algebraic geometry. Invent. Math. 127(3), 601-617 (1997) · Zbl 0909.14007
[4] K. Behrend, B. Fantechi, The intrinsic normal cone. Invent. Math. 128, 45-88 (1997) · Zbl 0909.14006
[5] B. Bertrand, E. Brugallé, G. Mikhalkin, Tropical open Hurwitz numbers. Rend. Semin. Mat. Univ. Padova 125, 157-171 (2011) · Zbl 1226.14066
[6] B. Bertrand, E. Brugallé, G. Mikhalkin, Genus 0 characteristic numbers of tropical projective plane. Compos. Math. 150(1), 46-104 (2014). arXiv:1105.2004
[7] E. Brugallé, H. Markwig, Deformation of tropical Hirzebruch surfaces and enumerative geometry. J. Algebraic Geom. (2013, preprint). arXiv:1303.1340 · Zbl 1396.14065
[8] A. Buchholz, H. Markwig, Tropical covers of curves and their moduli spaces. Commun. Contemp. Math. (2013). https://doi.org/10.1142/S0219199713500454 · Zbl 1318.14060
[9] L. Caporaso, Gonality of algebraic curves and graphs. Springer Proc. Math. Stat. 71, 77-108 (2014) · Zbl 1395.14026
[10] R. Cavalieri, P. Johnson, H. Markwig, Tropical Hurwitz numbers. J. Algebr. Comb. 32(2), 241-265 (2010). arXiv:0804.0579. https://doi.org/10.1007/s10801-009-0213-0 · Zbl 1218.14058
[11] R. Cavalieri, P. Johnson, H. Markwig, Wall crossings for double Hurwitz numbers. Adv. Math. 228(4), 1894-1937 (2011). arXiv:1003.1805 · Zbl 1231.14023
[12] R. Cavalieri, H. Markwig, D. Ranganathan, Tropicalizing the space of admissible covers. Math. Ann. (2014, preprint). arXiv:1401.4626 · Zbl 1373.14064
[13] R. Cavalieri, H. Markwig, D. Ranganathan, Tropical compactification and the Gromov-Witten theory of \(\(\mathbb {P}^1\)\). Sel. Math. 1(34) (2016). arXiv:1410.2837. https://doi.org/10.1007/s00029-016-0265-7 · Zbl 1391.14111
[14] E.M. Feichtner, B. Sturmfels, Matroid polytopes, nested sets and Bergman fans. Port. Math. 62, 437-468 (2005). arXiv:math.CO:0411260 · Zbl 1092.52006
[15] W. Fulton, R. Pandharipande, Notes on stable maps and quantum cohomology, in \(Algebraic Geometry: Santa Cruz 1995\), ed. by J. Kollár et al. Proceedings of Symposia in Pure Mathematics, vol. 62(2) (American Mathematical Society, Providence, RI, 1997), pp. 45-96 · Zbl 0898.14018
[16] A. Gathmann, D. Ochse, Moduli spaces of curves in tropical varieties, in \(Algorithmic and Experimental Methods in Algebra Geometry, and Number Theory\), ed. by G. Böckle, W. Decker, G. Malle (Springer, Heidelberg, 2018). https://doi.org/10.1007/978-3-319-70566-8_11
[17] A. Gathmann, M. Kerber, H. Markwig, Tropical fans and the moduli space of rational tropical curves. Compos. Math. 145(1), 173-195 (2009). arXiv:0708.2268 · Zbl 1169.51021
[18] E. Gawrilow, M. Joswig, Polymake: a framework for analyzing convex polytopes, in \(Polytopes - Combinatorics and Computation\), ed. by G. Kalai, G.M. Ziegler (Birkhäuser, Boston, 2000), pp. 43-74 · Zbl 0960.68182
[19] A. Gross, Correspondence theorems via tropicalizations of moduli spaces. Commun. Contemp. Math. (2014, to appear). arXiv:1406.1999 · Zbl 1387.14152
[20] S. Hampe, a-tint: a polymake extension for algorithmic tropical intersection theory. Eur. J. Comb. 36C, 579-607 (2014). arXiv:1208.4248 · Zbl 1285.14071
[21] H. Markwig, J. Rau, Tropical descendant Gromov-Witten invariants. Manuscripta Math. 129(3), 293-335 (2009). arXiv:0809.1102. https://doi.org/10.1007/s00229-009-0256-5 · Zbl 1171.14039
[22] G. Mikhalkin, Enumerative tropical geometry in \(\({\mathbb {R}^2}\)\). J. Am. Math. Soc. 18, 313-377 (2005). arXiv:math.AG/0312530 · Zbl 1092.14068
[23] G. Mikhalkin, Moduli spaces of rational tropical curves, in \(Proceedings of Gökova Geometry-Topology Conference GGT 2006\), pp. 39-51 (2007). arXiv:0704.0839 · Zbl 1203.14027
[24] D. Ochse, Moduli spaces of rational tropical stable maps into smooth tropical varieties. Ph.D. thesis, TU Kaiserslautern (2013)
[25] D. Speyer, B. Sturmfels, The tropical Grassmannian. Adv. Geom. 4, 389-411 (2004) · Zbl 1065.14071
[26] The GAP Group: GAP - Groups, Algorithms, and Programming, Version 4.8.6 (2016). http://www.gap-system.org
[27] R. Vakil, The enumerative geometry of rational and elliptic curves in projective space. J. Reine Angew. Math. (Crelle’s Journal) 529, 101-153 (2000) · Zbl 0970.14029
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