Geometry & Topology, Vol. 9 (2005) Paper no. 14, pages 483--491.

Logarithmic asymptotics of the genus zero Gromov-Witten invariants of the blown up plane

Ilia Itenberg, Viatcheslav Kharlamov and Eugenii Shustin

Abstract. We study the growth of the genus zero Gromov-Witten invariants GW_{nD} of the projective plane P^2_k blown up at k points (where D is a class in the second homology group of P^2_k). We prove that, under some natural restrictions on D, the sequence log GW_{nD} is equivalent to lambda n log n, where lambda = D.c_1(P^2_k).

Keywords. Gromov-Witten invariants, rational and ruled algebraic surfaces, rational and ruled symplectic 4-manifolds, tropical enumerative geometry

AMS subject classification. Primary: 14N35. Secondary: 14J26, 53D45.

DOI: 10.2140/gt.2005.9.483

E-print: arXiv:math.AG/0412533

Submitted to GT on 30 December 2004. Paper accepted 25 March 2005. Paper published 7 April 2005.

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Ilia Itenberg, Viatcheslav Kharlamov and Eugenii Shustin
(II and VK) Universite Louis Pasteur et IRMA, 7, rue Rene Descartes
67084 Strasbourg Cedex, France
and
(ES) School of Mathematical Sciences, Tel Aviv University
Ramat Aviv, 69978 Tel Aviv, Israel
Email: itenberg@math.u-strasbg.fr, kharlam@math.u-strasbg.fr, shustin@post.tau.ac.il

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