Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)



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
Published electronically: January 20, 2005
MathSciNet review: 2137980
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. O. Aharony, A. Hanany, Branes, superpotentials and superconformal fixed points, Nuclear Phys. B 504 (1997), 239-271. MR 1482482 (99c:81225)
  • 2. O. Aharony, A. Hanany, B. Kol, Webs of $(p,q)$ 5-branes, five dimensional field theories and grid diagrams, hep-th/9710116.
  • 3. L. Caporaso, J. Harris, Counting plane curves of any genus, Invent. Math. 131 (1998), 345-392. MR 1608583 (99i:14064)
  • 4. Y. Eliashberg, A. Givental, H. Hofer, Introduction to symplectic field theory, GAFA 2000, Special Volume, Part II, 560-673. MR 1826267 (2002e:53136)
  • 5. M. Forsberg, M. Passare, A. Tsikh, Laurent determinants and arangements of hyperplane amoebas, Advances in Math. 151 (2000), 45-70. MR 1752241 (2001m:32060)
  • 6. I. M. Gelfand, M. M. Kapranov, A. V. Zelevinski, Discriminants, resultants and multidimensional determinants, Birkhäuser, Boston, 1994. MR 1264417 (95e:14045)
  • 7. J. Harris, D. Morrison, Moduli of curves, Graduate Texts in Mathematics, 187, Springer-Verlag, New York, 1998. MR 1631825 (99g:14031)
  • 8. I. Itenberg, Amibes de variétés algébriques et dénombrement de courbes [d'après G. Mikhalkin], Séminaire Bourbaki 55ème année, 2002-2003, no. 921.
  • 9. I. Itenberg, V. Kharlamov, E. Shustin, Welschinger invariant and enumeration of real plane rational curves, Int. Math. Res. Not. 2003, no. 49, 2639-2653. MR 2012521 (2004h:14065)
  • 10. M. M. Kapranov, Amoebas over non-Archimedian fields, Preprint 2000.
  • 11. A. G. Khovanskii, Newton polyhedra and toric varieties (in Russian), Funkcional. Anal. i Prilozen. 11 (1977), no. 4, 56-64. MR 0476733 (57:16291)
  • 12. M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology and enumerative geometry, Comm. Math. Phys. 164 (1994), 525-562. MR 1291244 (95i:14049)
  • 13. M. Kontsevich, Y. Soibelman, Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000), 203-263, World Sci. Publishing, River Edge, NJ, 2001. MR 1882331 (2003c:32025)
  • 14. A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1-31. MR 0419433 (54:7454)
  • 15. G.L. Litvinov and V.P. Maslov, The correspondence principle for idempotent calculus and some computer applications, in Idempotency, J. Gunawardena (Editor), Cambridge Univ. Press, Cambridge, 1998, 420-443. MR 1608383 (99c:16050)
  • 16. G. Mikhalkin, Real algebraic curves, moment map and amoebas, Ann. of Math. 151 (2000), 309-326. MR 1745011 (2001c:14083)
  • 17. G. Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology 43 (2004), 1035-1065. MR 2079993
  • 18. G. Mikhalkin, Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629-634. MR 1988122 (2004d:14077)
  • 19. M. Passare, H. Rullgård, Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope. Duke Math. J. 121 (2004), 481-507. MR 2040284 (2005a:32005)
  • 20. J.-E. Pin, Tropical semirings, Idempotency (Bristol, 1994), 50-69, Publ. Newton Inst., 11, Cambridge Univ. Press, Cambridge, 1998. MR 1608374 (99a:16045)
  • 21. E. Shustin, Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry, math.AG/0211278.
  • 22. Z. Ran, Enumerative geometry of singular plane curves, Invent. Math. 97 (1989), 447-465. MR 1005002 (90g:14039)
  • 23. J. Richter-Gebert, B. Sturmfels, T. Theobald, First steps in tropical geometry, math.AG/0306366.
  • 24. D. Speyer, B. Sturmfels, The tropical Grassmanian, Adv. Geom. 4 (2004), 389-411. MR 2071813
  • 25. D. Speyer, B. Sturmfels, Tropical Mathematics, math.CO/0408099.
  • 26. A. Strominger, S.-T. Yau, E. Zaslow, Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243-259. MR 1429831 (97j:32022)
  • 27. B. Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, AMS, Providence, RI, 2002. MR 1925796 (2003i:13037)
  • 28. R. Vakil, Counting curves on rational surfaces, Manuscripta Math. 102 (2000), 53-84. MR 1771228 (2001h:14069)
  • 29. O.Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees $6$ and $7$, Lecture Notes in Math. 1060 (1984), 187-200, see also a preprint with a more modern and detailed presentation available at MR 0770238 (87i:14029)
  • 30. J.-Y. 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. MR 1976315 (2004m:53157)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 14N35, 52B20, 14N10, 14P25, 51M20

Retrieve articles in all journals with MSC (2000): 14N35, 52B20, 14N10, 14P25, 51M20

Additional Information

Grigory Mikhalkin
Affiliation: Department of Mathematics, University of Toronto, 100 St. George St., Toronto, Ontario, M5S 3G3 Canada and St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011 Russia
Address at time of publication: IHES, Le Bois-Marie, 35, route de Chartres, Bures-sur-Yvette, 91440, France

PII: S 0894-0347(05)00477-7
Keywords: Tropical curves, enumerative geometry, Gromov-Witten invariants, toric surfaces
Received by editor(s): January 31, 2004
Published electronically: January 20, 2005
Additional Notes: The author would like to acknowledge partial support of the NSF and NSERC
Article copyright: © Copyright 2005 Grigory Mikhalkin