#### 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.

Notes on file formats
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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