Enumerative tropical algebraic geometry in ℝ²

Mikhalkin, Grigory

doi:10.1090/S0894-0347-05-00477-7

Enumerative tropical algebraic geometry in $\mathbb{R} ^2$

Author: Grigory Mikhalkin
Journal: J. Amer. Math. Soc. 18 (2005), 313-377
MSC (2000): Primary 14N35, 52B20; Secondary 14N10, 14P25, 51M20
DOI: https://doi.org/10.1090/S0894-0347-05-00477-7
Published electronically: January 20, 2005
MathSciNet review: 2137980
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Abstract: The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629-634.

The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the real space $\mathbb{R} ^n$ and holomorphic curves with certain piecewise-linear graphs there.

1. Ofer Aharony and Amihay Hanany, Branes, superpotentials and superconformal fixed points, Nuclear Phys. B 504 (1997), no. 1-2, 239–271. MR 1482482, https://doi.org/10.1016/S0550-3213(97)00472-0
2. O. Aharony, A. Hanany, B. Kol, Webs of 5-branes, five dimensional field theories and grid diagrams, http://arxiv.org hep-th/9710116.
3. Lucia Caporaso and Joe Harris, Counting plane curves of any genus, Invent. Math. 131 (1998), no. 2, 345–392. MR 1608583, https://doi.org/10.1007/s002220050208
4. Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume (2000), 560–673. GAFA 2000 (Tel Aviv, 1999). MR 1826267, https://doi.org/10.1007/978-3-0346-0425-3_4
5. Mikael Forsberg, Mikael Passare, and August Tsikh, Laurent determinants and arrangements of hyperplane amoebas, Adv. Math. 151 (2000), no. 1, 45–70. MR 1752241, https://doi.org/10.1006/aima.1999.1856
6. I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417
7. Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR 1631825
8. I. Itenberg, Amibes de variétés algébriques et dénombrement de courbes [d'après G. Mikhalkin], Séminaire Bourbaki 55ème année, 2002-2003, no. 921.
9. Ilia Itenberg, Viatcheslav Kharlamov, and Eugenii Shustin, Welschinger invariant and enumeration of real rational curves, Int. Math. Res. Not. 49 (2003), 2639–2653. MR 2012521, https://doi.org/10.1155/S1073792803131352
10. M. M. Kapranov, Amoebas over non-Archimedian fields, Preprint 2000.
11. A. G. Hovanskiĭ, Newton polyhedra, and toroidal varieties, Funkcional. Anal. i Priložen. 11 (1977), no. 4, 56–64, 96 (Russian). MR 0476733
12. M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR 1291244
13. Maxim Kontsevich and Yan Soibelman, Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000) World Sci. Publ., River Edge, NJ, 2001, pp. 203–263. MR 1882331, https://doi.org/10.1142/9789812799821_0007
14. A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1–31 (French). MR 419433, https://doi.org/10.1007/BF01389769
15. Grigori L. Litvinov and Victor P. Maslov, The correspondence principle for idempotent calculus and some computer applications, Idempotency (Bristol, 1994) Publ. Newton Inst., vol. 11, Cambridge Univ. Press, Cambridge, 1998, pp. 420–443. MR 1608383, https://doi.org/10.1017/CBO9780511662508.026
16. G. Mikhalkin, Real algebraic curves, the moment map and amoebas, Ann. of Math. (2) 151 (2000), no. 1, 309–326. MR 1745011, https://doi.org/10.2307/121119
17. Grigory Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology 43 (2004), no. 5, 1035–1065. MR 2079993, https://doi.org/10.1016/j.top.2003.11.006
18. Grigory Mikhalkin, Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629–634 (English, with English and French summaries). MR 1988122, https://doi.org/10.1016/S1631-073X(03)00104-3
19. Mikael Passare and Hans Rullgård, Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope, Duke Math. J. 121 (2004), no. 3, 481–507. MR 2040284, https://doi.org/10.1215/S0012-7094-04-12134-7
20. Jean-Eric Pin, Tropical semirings, Idempotency (Bristol, 1994) Publ. Newton Inst., vol. 11, Cambridge Univ. Press, Cambridge, 1998, pp. 50–69. MR 1608374, https://doi.org/10.1017/CBO9780511662508.004
21. E. Shustin, Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry, http://arxiv.org math.AG/0211278.
22. Z. Ran, Enumerative geometry of singular plane curves, Invent. Math. 97 (1989), no. 3, 447–465. MR 1005002, https://doi.org/10.1007/BF01388886
23. J. Richter-Gebert, B. Sturmfels, T. Theobald, First steps in tropical geometry, http://arXiv.org math.AG/0306366.
24. David Speyer and Bernd Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004), no. 3, 389–411. MR 2071813, https://doi.org/10.1515/advg.2004.023
25. D. Speyer, B. Sturmfels, Tropical Mathematics, http://arxiv.org math.CO/0408099.
26. Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, Mirror symmetry is 𝑇-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243–259. MR 1429831, https://doi.org/10.1016/0550-3213(96)00434-8
27. Bernd Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, vol. 97, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2002. MR 1925796
28. Ravi Vakil, Counting curves on rational surfaces, Manuscripta Math. 102 (2000), no. 1, 53–84. MR 1771228, https://doi.org/10.1007/s002291020053
29. O. Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Topology (Leningrad, 1982) Lecture Notes in Math., vol. 1060, Springer, Berlin, 1984, pp. 187–200. MR 770238, https://doi.org/10.1007/BFb0099934
30. Jean-Yves Welschinger, Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry, C. R. Math. Acad. Sci. Paris 336 (2003), no. 4, 341–344 (English, with English and French summaries). MR 1976315, https://doi.org/10.1016/S1631-073X(03)00059-1