Geometry & Topology Volume 26, issue 8 (2022)

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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Tropical $\psi$ classes

Renzo Cavalieri, Andreas Gross and Hannah Markwig

Geometry & Topology 26 (2022) 3421–3524

DOI: 10.2140/gt.2022.26.3421

Bibliography

1	D Abramovich, L Caporaso, S Payne, The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér. 48 (2015) 765 MR3377065
2	L Allermann, J Rau, First steps in tropical intersection theory, Math. Z. 264 (2010) 633 MR2591823
3	E Arbarello, M Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5 (1996) 705 MR1486986
4	P Belmans, A J de Jong, others, The Stacks project, electronic reference (2005–)
5	B Bertrand, E Brugallé, G Mikhalkin, Tropical open Hurwitz numbers, Rend. Semin. Mat. Univ. Padova 125 (2011) 157 MR2866125
6	L J Billera, S P Holmes, K Vogtmann, Geometry of the space of phylogenetic trees, Adv. in Appl. Math. 27 (2001) 733 MR1867931
7	E Brugallé, I Itenberg, G Mikhalkin, K Shaw, Brief introduction to tropical geometry, from: "Proceedings of the Gökova Geometry–Topology Conference 2014" (editors S Akbulut, D Auroux, T Önder), GGT (2015) 1 MR3381439
8	E Brugallé, G Mikhalkin, Floor decompositions of tropical curves: the planar case, from: "Proceedings of Gökova Geometry–Topology Conference 2008" (editors S Akbulut, T Önder, R J Stern), GGT (2009) 64 MR2500574
9	L Caporaso, Algebraic and tropical curves: comparing their moduli spaces, from: "Handbook of moduli, I" (editors G Farkas, I Morrison), Adv. Lect. Math. 24, International (2013) 119 MR3184163
10	L Caporaso, Gonality of algebraic curves and graphs, from: "Algebraic and complex geometry" (editors A Frühbis-Krüger, R N Kloosterman, M Schütt), Springer Proc. Math. Stat. 71, Springer (2014) 77 MR3278571
11	L Caporaso, Recursive combinatorial aspects of compactified moduli spaces, from: "Proceedings of the International Congress of Mathematicians" (editors B Sirakov, P N d Souza, M Viana), World Sci. (2018) 635 MR3966783
12	D Cartwright, Combinatorial tropical surfaces, preprint (2015) arXiv:1506.02023
13	R Cavalieri, M Chan, M Ulirsch, J Wise, A moduli stack of tropical curves, Forum Math. Sigma 8 (2020) MR4091085
14	R Cavalieri, H Markwig, D Ranganathan, Tropicalizing the space of admissible covers, Math. Ann. 364 (2016) 1275 MR3466867
15	S Flossmann, Tropical intersection theory on ℝⁿ, preprint (2014) arXiv:1405.5018
16	S Fomin, G Mikhalkin, Labeled floor diagrams for plane curves, J. Eur. Math. Soc. 12 (2010) 1453 MR2734349
17	G François, J Rau, The diagonal of tropical matroid varieties and cycle intersections, Collect. Math. 64 (2013) 185 MR3041763
18	G Francois, S Hampe, Universal families of rational tropical curves, Canad. J. Math. 65 (2013) 120 MR3004460
19	W Fulton, B Sturmfels, Intersection theory on toric varieties, Topology 36 (1997) 335 MR1415592
20	A Gathmann, M Kerber, H Markwig, Tropical fans and the moduli spaces of tropical curves, Compos. Math. 145 (2009) 173 MR2480499
21	A Gathmann, H Markwig, Kontsevich’s formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008) 537 MR2370275
22	E Gawrilow, M Joswig, polymake: a framework for analyzing convex polytopes, from: "Polytopes: combinatorics and computation" (editors G Kalai, G M Ziegler), DMV Sem. 29, Birkhäuser (2000) 43 MR1785292
23	A Gibney, D Maclagan, Equations for Chow and Hilbert quotients, Algebra Number Theory 4 (2010) 855 MR2776876
24	C Goldner, Counting tropical rational curves with cross-ratio constraints, Math. Z. 297 (2021) 133 MR4204688
25	A Gross, Correspondence theorems via tropicalizations of moduli spaces, Commun. Contemp. Math. 18 (2016) MR3477404
26	A Gross, Intersection theory on tropicalizations of toroidal embeddings, Proc. Lond. Math. Soc. 116 (2018) 1365 MR3816384
27	A Gross, F Shokrieh, A sheaf-theoretic approach to tropical homology, preprint (2019) arXiv:1906.09245
28	M Gross, B Siebert, Mirror symmetry via logarithmic degeneration data, I, J. Differential Geom. 72 (2006) 169 MR2213573
29	S Hampe, a-tint : a polymake extension for algorithmic tropical intersection theory, European J. Combin. 36 (2014) 579 MR3131916
30	I Itenberg, L Katzarkov, G Mikhalkin, I Zharkov, Tropical homology, Math. Ann. 374 (2019) 963 MR3961331
31	P Jell, J Rau, K Shaw, Lefschetz (1,1)–theorem in tropical geometry, Épijournal Géom. Algébrique 2 (2018) MR3894860
32	Z. Jin, Tropical intersection theory on moduli stack of curve coverings, PhD thesis, University of Cambridge (2019)
33	M M Kapranov, Chow quotients of Grassmannians, I, from: "I M Gelfand Seminar" (editors S Gelfand, S Gindikin), Adv. Soviet Math. 16, Amer. Math. Soc. (1993) 29 MR1237834
34	E Katz, Tropical intersection theory from toric varieties, Collect. Math. 63 (2012) 29 MR2887109
35	M Kerber, H Markwig, Counting tropical elliptic plane curves with fixed j–invariant, Comment. Math. Helv. 84 (2009) 387 MR2495799
36	M Kerber, H Markwig, Intersecting Psi-classes on tropical ℳ_0,n, Int. Math. Res. Not. 2009 (2009) 221 MR2482115
37	M Kontsevich, Enumeration of rational curves via torus actions, from: "The moduli space of curves" (editors R Dijkgraaf, C Faber, G van der Geer), Progr. Math. 129, Birkhäuser (1995) 335 MR1363062
38	G Mikhalkin, Enumerative tropical algebraic geometry in ℝ², J. Amer. Math. Soc. 18 (2005) 313 MR2137980
39	G Mikhalkin, Tropical geometry and its applications, from: "International Congress of Mathematicians" (editors B Sirakov, P N d Souza, M Viana), Eur. Math. Soc. (2006) 827
40	G Mikhalkin, Moduli spaces of rational tropical curves, from: "Proceedings of Gökova Geometry–Topology Conference 2006" (editors S Akbulut, T Önder, R J Stern), GGT (2007) 39
41	G. Mikhalkin, J. Rau, Tropical geometry, book project (2020)
42	G Mikhalkin, I Zharkov, Tropical curves, their Jacobians and theta functions, from: "Curves and abelian varieties" (editors V Alexeev, A Beauville, C H Clemens, E Izadi), Contemp. Math. 465, Amer. Math. Soc. (2008) 203 MR2457739
43	G Mikhalkin, I Zharkov, Tropical eigenwave and intermediate Jacobians, from: "Homological mirror symmetry and tropical geometry" (editors R Castaño-Bernard, F Catanese, M Kontsevich, T Pantev, Y Soibelman, I Zharkov), Lect. Notes Unione Mat. Ital. 15, Springer (2014) 309 MR3330789
44	D Mumford, Towards an enumerative geometry of the moduli space of curves, from: "Arithmetic and geometry, II" (editors M Artin, J Tate), Progr. Math. 36, Birkhäuser (1983) 271 MR717614
45	D Ranganathan, Skeletons of stable maps, II : superabundant geometries, Res. Math. Sci. 4 (2017) MR3657349
46	D Ranganathan, K Santos-Parker, J Wise, Moduli of stable maps in genus one and logarithmic geometry, I, Geom. Topol. 23 (2019) 3315 MR4046967
47	D Ranganathan, K Santos-Parker, J Wise, Moduli of stable maps in genus one and logarithmic geometry, II, Algebra Number Theory 13 (2019) 1765 MR4017534
48	J Rau, Intersections on tropical moduli spaces, Rocky Mountain J. Math. 46 (2016) 581 MR3529085
49	K M Shaw, A tropical intersection product in matroidal fans, SIAM J. Discrete Math. 27 (2013) 459 MR3032930
50	K Shaw, Tropical surfaces, preprint (2015) arXiv:1506.07407
51	D E Speyer, Tropical geometry, PhD thesis, University of California, Berkeley (2005) MR2707751
52	D E Speyer, Parameterizing tropical curves, I : Curves of genus zero and one, Algebra Number Theory 8 (2014) 963 MR3248991
53	D Speyer, B Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004) 389 MR2071813
54	J Tevelev, Compactifications of subvarieties of tori, Amer. J. Math. 129 (2007) 1087 MR2343384
55	C. Torchiani, Enumerative geometry of rational and elliptic tropical curves in ℝ^m, PhD thesis, Universität Kaiserslautern (2014)
56	I Tyomkin, Tropical geometry and correspondence theorems via toric stacks, Math. Ann. 353 (2012) 945 MR2923954
57	I Tyomkin, Enumeration of rational curves with cross-ratio constraints, Adv. Math. 305 (2017) 1356 MR3570161
58	M Ulirsch, Functorial tropicalization of logarithmic schemes: the case of constant coefficients, Proc. Lond. Math. Soc. 114 (2017) 1081 MR3661346
59	M Ulirsch, A non-Archimedean analogue of Teichmüller space and its tropicalization, Selecta Math. (N.S.) 27 (2021) MR4265944
60	R Vakil, The moduli space of curves and Gromov–Witten theory, from: "Enumerative invariants in algebraic geometry and string theory" (editors K Behrend, M Manetti), Lecture Notes in Math. 1947, Springer (2008) 143 MR2493586
61	E Witten, Two-dimensional gravity and intersection theory on moduli space, from: "Surveys in differential geometry" (editors H B Lawson Jr., S T Yau), Lehigh Univ. (1991) 243 MR1144529
62	B Zimmermann, Finite groups of outer automorphisms of free groups, Glasgow Math. J. 38 (1996) 275 MR1417356