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Tropical compactification and the Gromov-Witten theory of $$\mathbb {P}^1$$. (English) Zbl 1391.14111
Tropical geometry is a systematic method to study algebraic geometry by using piecewise-linear shadows of solutions to polynomial equations. In particular, it provides combinatorially effective tools to approach various enumerative problems in classical algebraic geometry. The first important result in this field is the Tropical Correspodence Theorem of G. Mikhalkin [J. Am. Math. Soc. 18, No. 2, 313–377 (2005; Zbl 1092.14068)], between algebraic curves in $$\mathbb{P}^2$$ and tropical curves in $$\mathbb{R}^2$$. Later on, this correspondence was generalized also to higher dimensional toric varieties by T. Nishinou and B. Siebert [Duke Math. J. 135, No. 1, 1–51 (2006; Zbl 1105.14073)].
The paper under review establishes a new tropical correspondence theorem for genus $$0$$ stable maps to $$\mathbb{P}^1$$ relative to two points $$\{ 0,\infty \} \in \mathbb{P}^1$$. The significance of the result is that it is established on the level of moduli spaces, providing a deeper connection between the classical and tropical moduli spaces for such maps.
Let $$\mathcal{M}(x^{+},x^{-})$$ denote the moduli space of genus $$0$$ stable maps to $$\mathbb{P}^1$$, with ramification profiles over $$0$$ and $$\infty$$ specified by tuples of integers $$x^{+} = (x_1,\ldots,x_l) \in \mathbb{Z}^l_{> 0}$$ and $$x^{-}=(x_{l+1},\ldots,x_{n}) \in \mathbb{Z}^{n-l}_{< 0}$$ respectively, with the additional assumption that $$\sum_{i=1}^{n} x_i=0$$. In the article it is shown that there is an embedding $$\mathcal{M}(x^{+},x^{-}) \hookrightarrow T$$, where $$T$$ is the dense torus orbit of the toric variety $$X_{\sigma}$$ associated to the fan $$\sigma$$ defined by the tropical moduli space $$\mathcal{\overline{M}^{\mathrm{trop}}}(x^{+},x^{-})$$, such that the coarse moduli space $$\mathcal{\overline{M}}(x^{+},x^{-})$$ is identified with the tropical compactification of the image of $$\mathcal{M}(x^{+},x^{-})$$ in $$X_{\sigma}$$. This result is build on various observations, in particular on results of the two papers [M. M. Kapranov, in: I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in functional analysis held at Moscow University, Russia, September 1993. Providence, RI: American Mathematical Society. 29–110 (1993; Zbl 0811.14043)] and [A. Gibney and D. Maclagan, Algebra Number Theory 4, No. 7, 855–885 (2010; Zbl 1210.14051)]. First, from the article of Kapranov, it follows that moduli space $$\mathcal{M}(0,n)$$ of genus zero stable maps with $$n$$-marked points can be embedded into a toric variety X, given by a Chow quetient where the closure of the image gives the compactification $$\mathcal{\overline{M}}(0,n)$$. Second, Gibney and Maclagan show that the tropicalization of the moduli space of stable maps under this embedding is given by a subfan of the toric fan for $$X$$. The observation that the natural forgetful between $$\mathcal{\overline{M}}(x^{+},x^{-})$$ and $$\mathcal{\overline{M}}(0,n)$$ induces an isomorphism over the open set $$\mathcal{M}(x^{+},x^{-})$$ leads us to an embedding of $$\mathcal{M}(x^{+},x^{-}) \hookrightarrow T$$, where $$T$$ is the dense torus orbit of the toric variety associated to the fan given by $$\mathcal{\overline{M}}^{\mathrm{trop}}(x^{+},x^{-})$$. In the article it is then shown that the closure of $$\mathcal{M}(x^{+},x^{-})$$ under this embedding can be identified with the coarse moduli space $$\mathcal{\overline{M}}(x^{+},x^{-})$$. As an application of this result, a correspondence between tropical Hurwitz cycles and tropicalizations of classical Hurwitz cycles, as well as a tropical descendant correspondence for genus $$0$$ relative invariants of $$\mathbb{P}^1$$ is established.

##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14T05 Tropical geometry (MSC2010)
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##### References:
 [1] Abramovich, D; Caporaso, L; Payne, S, The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér., 48, 765-809, (2015) · Zbl 1410.14049 [2] Abramovich, D; Chen, Q, Stable logarithmic maps to Deligne-faltings pairs. II, Asian J. Math, 18, 465-488, (2014) · Zbl 1321.14025 [3] Abramovich, D., Chen, Q., Marcus, S., Ulirsch, M., Wise, J.:Skeletons and Fans of Logarithmic Structures. arXiv:1503.04343 (2015) · Zbl 1364.14047 [4] Abramovich, D; Corti, A; Vistoli, A, Twisted bundles and admissible covers, Commun. Algebra, 31, 3547-3618, (2003) · Zbl 1077.14034 [5] Abramovich, D; Karu, K, Weak semistable reduction in characteristic 0, Invent. Math., 139, 241-273, (2000) · Zbl 0958.14006 [6] Abramovich, D; Marcus, S; Wise, J, Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations, Ann. Inst. Fourier, 64, 1611-1667, (2014) · Zbl 1317.14123 [7] Bertram, A; Cavalieri, R; Markwig, H, Polynomiality, wall crossings and tropical geometry of rational double Hurwitz cycles, J. Comb. Theory Ser. A, 120, 1604-1631, (2013) · Zbl 1317.14136 [8] Bertrand, B; Brugallé, E; Mikhalkin, G, Tropical open Hurwitz numbers, Rend. Semin. Mat. Univ. Padova, 125, 157-171, (2011) · Zbl 1226.14066 [9] Billera, L; Holmes, S; Vogtmann, K, Geometry of the space of phylogenetic trees, Adv. Appl. Math., 27, 733-767, (2001) · Zbl 0995.92035 [10] Cavalieri, R; Hampe, S; Markwig, H; Ranganathan, D, Moduli spaces of rational weighted stable curves and tropical geometry, Forum Math. Sigma, 4, e9, (2016) · Zbl 1373.14063 [11] Cavalieri, R; Johnson, P; Markwig, H, Tropical Hurwitz numbers, J. Algebraic Comb., 32, 241-265, (2010) · Zbl 1218.14058 [12] Cavalieri, R; Markwig, H; Ranganathan, D, Tropicalizing the space of admissible covers, Math. Ann., 364, 1275-1313, (2016) · Zbl 1373.14064 [13] Chen, Q, Stable logarithmic maps to Deligne-faltings pairs I, Ann. Math., 180, 455-521, (2014) · Zbl 1311.14028 [14] Fulton, W.: Introduction to Toric Varieties. The 1989 William H. Roever lectures in geometry. Princeton, NJ: Princeton University Press (1993) · Zbl 0813.14039 [15] Fulton, W; Sturmfels, B, Intersection theory on toric varieties, Topology, 36, 335-353, (1997) · Zbl 0885.14025 [16] Gathmann, A; Kerber, M; Markwig, H, Tropical fans and the moduli space of rational tropical curves, Comp. Math., 145, 173-195, (2009) · Zbl 1169.51021 [17] Gibney, A; Maclagan, D, Equations for Chow and Hilbert quotients, Algebra Number Theory J., 4, 855-885, (2010) · Zbl 1210.14051 [18] Goulden, I; Jackson, D; Vakil, R, Towards the geometry of double Hurwitz numbers, Adv. Math., 198, 43-92, (2005) · Zbl 1086.14022 [19] Graber, T; Vakil, R, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math. J., 130, 1-37, (2005) · Zbl 1088.14007 [20] Gross, A: Correspondence Theorems via Tropicalizations of Moduli Spaces. arXiv:1401.4626 (2014) · Zbl 1387.14152 [21] Gross, A.: Intersection Theory on Tropicalizations of Toroidal Embeddings. arXiv:1510.04604 (2015) · Zbl 1420.14142 [22] Gross, M, Mirror symmetry for $$\mathbb{P}^2$$ and tropical geometry, Adv. Math., 224, 169-245, (2010) · Zbl 1190.14038 [23] Gross, M; Siebert, B, Logarithmic Gromov-Witten invariants, J. Am. Math. Soc., 26, 451-510, (2013) · Zbl 1281.14044 [24] Gubler, W.: A guide to tropicalizations. In: Algebraic and Combinatorial Aspects of Tropical Geometry, vol. 589 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2013, pp. 125-189 · Zbl 1318.14061 [25] Hampe, S.: Combinatorics of Tropical Hurwitz Cycles. arXiv:1407.3933 (2014) · Zbl 1335.14015 [26] Kapranov, M, Chow quotients of Grassmannians I, IM Gelfand Semin., 16, 29-110, (1993) · Zbl 0811.14043 [27] Katz, E, Tropical intersection theory from toric varieties, Collect. Math., 63, 29-44, (2012) · Zbl 1322.14091 [28] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I, Lecture Notes in Mathematics, 339 (1973) · Zbl 0271.14017 [29] Kerber, M; Markwig, H, Intersecting psi-classes on tropical $$M_{0, n}$$, Int. Math. Res. Not., 2009, 221-240, (2009) · Zbl 1205.14070 [30] Maclagan, D., Sturmfels, B.: Introduction to tropical geometry, vol. 161, AMS Graduate Studies in Mathematics (2015) · Zbl 1321.14048 [31] Markwig, H; Rau, J, Tropical descendant Gromov-Witten invariants, Manuscr. Math., 129, 293-335, (2009) · Zbl 1171.14039 [32] Maulik, D; Pandharipande, R, A topological view of Gromov-Witten theory, Topology, 45, 887-918, (2006) · Zbl 1112.14065 [33] Mikhalkin, G, Enumerative tropical geometry in $$\mathbb{R}^2$$, J. Am. Math. Soc., 18, 313-377, (2005) · Zbl 1092.14068 [34] Mikhalkin, G.: Moduli spaces of rational tropical curves, pp. 39-51 (2007) · Zbl 1203.14027 [35] Okounkov, A; Pandharipande, R, Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. Math., 163, 517-560, (2006) · Zbl 1105.14076 [36] Osserman, B; Payne, S, Lifting tropical intersections, Doc. Math., 18, 121-175, (2013) · Zbl 1308.14069 [37] Osserman, B., Rabinoff, J.: Lifting non-proper tropical intersections. In: O. Amini, M. Baker, X. Faber (eds). Tropical and Non-archimedean Geometry, vol. 605, pp. 15-44 (2011) · Zbl 1320.14078 [38] Overholser, D. P.: Descendent Tropical Mirror Symmetry for $${\mathbb{P}}^2$$. arXiv:1504.06138 (2015) · Zbl 1415.14021 [39] Payne, S, Analytification is the limit of all tropicalizations, Math. Res. Lett., 16, 543-556, (2009) · Zbl 1193.14077 [40] Payne, S, Topology of Nonarchimedean analytic spaces and relations to complex algebraic geometry, Bull. Am. Math. Soc., 52, 223-247, (2015) · Zbl 1317.32046 [41] Ranganathan, D.: Moduli of Rational Curves in Toric Varieties and Non-archimedean Geometry. arXiv:1506.03754 (2015) [42] Speyer, D; Sturmfels, B, The tropical Grassmannian, Adv. Geom., 4, 389-411, (2004) · Zbl 1065.14071 [43] Tevelev, J, Compactifications of subvarieties of tori, Am. J. Math., 129, 1087-1104, (2007) · Zbl 1154.14039 [44] Thuillier, A, Géométrie toroïdale et géométrie analytique non archimédienne. application au type d’homotopie de certains schémas formels, Manuscr. Math., 123, 381-451, (2007) · Zbl 1134.14018 [45] Ulirsch, M.: Functorial Tropicalization of Logarithmic Schemes: The Case of Constant Coefficients. arXiv:1310.6269 (2013) · Zbl 1419.14088 [46] Ulirsch, M, Tropical geometry of moduli spaces of weighted stable curves, J. Lond. Math. Soc., 92, 427-450, (2015) · Zbl 1349.14197 [47] Vakil, R.: The moduli space of curves and Gromov-Witten theory. In: Enumerative Invariants in Algebraic Geometry and String Theory, pp. 143-198. Springer, Berlin (2008) · Zbl 1156.14043
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