A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces
Author:
Eugenii Shustin
Journal:
J. Algebraic Geom. 15 (2006), 285-322
DOI:
https://doi.org/10.1090/S1056-3911-06-00434-6
Published electronically:
January 11, 2006
MathSciNet review:
2199066
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References |
Additional Information
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.
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IKS2 I. Itenberg, V. Kharlamov, and E. Shustin, Logarithmic equivalence of Welschinger and Gromov-Witten invariants, Russian Math. Surveys 59 (2004), no. 6, 1093–1116.
K M. M. Kapranov, Amoebas over non-Archimedean fields, Preprint, 2000.
M G. Mikhalkin, Amoebas of algebraic varieties and tropical geometry, Different faces of geometry (Donaldson, S., et al., ed.) Kluwer, NY, 2004, pp. 257–300.
M2 G. Mikhalkin, Counting curves via the lattice paths in polygons, Comptes Rendus Math. 336 (2003), no. 8, 629–634.
M3 G. Mikhalkin, Enumerative tropical algebraic geometry in ${\mathbb R}^2$, J. Amer. Math. Soc. 18 (2005), 313–377.
N A. Nobile, On specialization of curves. I, Trans. Amer. Math. Soc. 282 (1984), no. 2, 739–748.
RST 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.
ShPo E. Shustin, Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry, Preprint arXiv:math.AG/0211278.
ShP E. Shustin, A tropical approach to enumerative geometry, Algebra i Analiz 17 (2005), no. 2, 170–214.
ST E. Shustin and I. Tyomkin, Patchworking singular algebraic curves. I, Israel Math. J., to appear.
St B. Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics. AMS, Providence, RI, 2002.
Wel 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.
Wel1 J.-Y. Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195–234.
Additional Information
Eugenii Shustin
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel
MR Author ID:
193452
Email:
shustin@post.tau.ac.il
Received by editor(s):
May 24, 2004
Received by editor(s) in revised form:
September 21, 2005
Published electronically:
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.
