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

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 $\lambda =D\cdot c_1\left(P_k^2\right)$.