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