Volume 9, issue 1 (2005)

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

Ilia Itenberg, Viatcheslav Kharlamov and Eugenii Shustin

Geometry & Topology 9 (2005) 483–491
 arXiv: math.AG/0412533
Abstract

We study the growth of the genus zero Gromov–Witten invariants $G{W}_{nD}$ of the projective plane ${P}_{k}^{2}$ blown up at $k$ points (where $D$ is a class in the second homology group of ${P}_{k}^{2}$). We prove that, under some natural restrictions on $D$, the sequence $logG{W}_{nD}$ is equivalent to $\lambda nlogn$, where $\lambda =D\cdot {c}_{1}\left({P}_{k}^{2}\right)$.

Keywords
Gromov–Witten invariants, rational, ruled algebraic surfaces, rational, ruled symplectic 4–manifolds, tropical enumerative geometry
Mathematical Subject Classification 2000
Primary: 14N35
Secondary: 14J26, 53D45
Publication
Received: 30 December 2004
Accepted: 25 March 2005
Published: 7 April 2005
Proposed: Yasha Eliashberg
Seconded: Leonid Polterovich, Simon Donaldson
Authors
 Ilia Itenberg Université Louis Pasteur et IRMA 7, rue René Descartes 67084 Strasbourg Cedex France Viatcheslav Kharlamov Université Louis Pasteur et IRMA 7, rue René Descartes 67084 Strasbourg Cedex France Eugenii Shustin School of Mathematical Sciences Tel Aviv University Ramat Aviv 69978 Tel Aviv Israel