Journal of Algebraic Geometry

A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces

Author(s): Eugenii Shustin
Journal: J. Algebraic Geom. 15 (2006), 285-322.
Posted: January 11, 2006
Retrieve article in: PDF DVI PostScript

Abstract: The Welschinger invariants of real rational algebraic surfaces are natural analogues of the genus zero Gromov-Witten invariants. We establish a tropical formula to calculate the Welschinger invariants of real toric Del Pezzo surfaces for any conjugation-invariant configuration of points. The formula expresses the Welschinger invariants via the total multiplicity of certain tropical curves (non-Archimedean amoebas) passing through generic configurations of points, and then via the total multiplicity of some lattice paths in the convex lattice polygon associated with a given surface. We also present the results of computation of Welschinger invariants, obtained jointly with I. Itenberg and V. Kharlamov.

1.: S. Diaz and J. Harris, Ideals associated to deformations of singular plane curves, Trans. Amer. Math. Soc. 309 (1988), no. 2, 433-468. MR 0961600 (89m:14003)
2.: I. Itenberg, Amibes des variétés algébriques et denombrement de courbes [d'après G. Mikhalkin], Séminaire N. Bourbaki 921, vol. 2002-03, Juin 2003.
3.: I. Itenberg, V. Kharlamov, and E. Shustin, Welschinger invariant and enumeration of real rational curves, Internat. Math. Res. Notices 49 (2003), 2639-2653. MR 2012521 (2004h:14065)
4.: I. Itenberg, V. Kharlamov, and E. Shustin, Appendix to ``Welschinger invariant and enumeration of real rational curves", Preprint arXiv:math/AG.0312142.
5.: I. Itenberg, V. Kharlamov, and E. Shustin, Logarithmic equivalence of Welschinger and Gromov-Witten invariants, Russian Math. Surveys 59 (2004), no. 6, 1093-1116. MR 2138469
6.: M. M. Kapranov, Amoebas over non-Archimedean fields, Preprint, 2000.
7.: G. Mikhalkin, Amoebas of algebraic varieties and tropical geometry, Different faces of geometry (Donaldson, S., et al., ed.) Kluwer, NY, 2004, pp. 257-300. MR 2102998 (2005m:14110)
8.: G. Mikhalkin, Counting curves via the lattice paths in polygons, Comptes Rendus Math. 336 (2003), no. 8, 629-634. MR 1988122 (2004d:14077)
9.: G. Mikhalkin, Enumerative tropical algebraic geometry in ${\mathbb{R}}^2$ , J. Amer. Math. Soc. 18 (2005), 313-377. MR 2137980
10.: A. Nobile, On specialization of curves. I, Trans. Amer. Math. Soc. 282 (1984), no. 2, 739-748. MR 0732116 (85j:14047)
11.: J. Richter-Gebert, B. Sturmfels, and T. Theobald, First steps in tropical geometry, Idempotent mathematics and mathematical physics, Contemp. Math., 377, Amer. Math. Soc., Providence, RI, 2005, pp. 289-317. MR 2149011
12.: E. Shustin, Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry, Preprint arXiv:math.AG/0211278.
13.: E. Shustin, A tropical approach to enumerative geometry, Algebra i Analiz 17 (2005), no. 2, 170-214. MR 2159589
14.: E. Shustin and I. Tyomkin, Patchworking singular algebraic curves. I, Israel Math. J., to appear.
15.: B. Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics. AMS, Providence, RI, 2002. MR 1925796 (2003i:13037)
16.: J.-Y. Welschinger, Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry, C. R. Acad. Sci. Paris, Sér. I 336 (2003), 341-344. MR 1976315 (2004m:53157)
17.: J.-Y. Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195-234.

Eugenii Shustin
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel
Email: shustin@post.tau.ac.il

PII: S 1056-3911(06)00434-6
Received by editor(s): May 24, 2004
Received by editor(s) in revised form: September 21, 2005
Posted: January 11, 2006
Additional Notes: Part of this work was done during the author's stay at Universität Kaiserslautern, supported by the Hermann-Minkowski Minerva Center for Geometry at Tel Aviv University, and during the author's stay at the Mathematical Science Research Institute, Berkeley. The author is very grateful to the Hermann-Minkowski Minerva Center for its support, and to Universität Kaiserslautern and MSRI for their hospitality and excellent working conditions.

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues