Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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

DOI: https://doi.org/10.1090/S1056-3911-06-00434-6
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.

American Mathematical Society