rational points on the unit circle. In this notation (IV) asserts that the chambers are disjoint.

[**6.**]{} Here is a sketch of a proof of (V) and (VI), which was worked out in a discussion with Yael Karshon. Construct a model for the symplectic blowup $M={{{{\mathbb C}}\mathrm{P}}}^2\#2\overline{{{{{\mathbb C}}\mathrm{P}}}}^2$ as follows. Choose distinct points $p_1,p_2\in{{{{\mathbb C}}\mathrm{P}}}^2\setminus{{{{\mathbb C}}\mathrm{P}}}^1$ and disjoint neighborhoods $U_1,U_2\subset{{{{\mathbb C}}\mathrm{P}}}^2\setminus{{{{\mathbb C}}\mathrm{P}}}^1$ of $p_1,p_2$, respectively, equipped with holomorphic coordinate charts $${{\varphi}}_i:B^4_{{\varepsilon}}\to U_i,$$ defined on the ${{\varepsilon}}$ ball in ${{\mathbb R}}^4={{\mathbb C}}^2$, centered at the origin. Then use the standard complex blowup construction at $p_1$ and $p_2$ to define $M$ as a complex manifold. There is a holomorphic projection $$\pi:M\to{{{{\mathbb C}}\mathrm{P}}}^2$$ with singular values $p_1,p_2$. Their preimages $S_1:=\pi^{-1}(p_1)$ and $S_2:=\pi^{-1}(p_2)$ are the exceptional divisors that, with their complex orientations, represent the classes $E_1,E_2$. The projection $\pi$ is a holomorphic diffeomorphism from $M\setminus(S_1\cup S_2)$ to ${{{{\mathbb C}}\mathrm{P}}}^2\setminus\{p_1,p_2\}$ and the holomorphic sphere $S:=\pi^{-1}({{{{\mathbb C}}\mathrm{P}}}^1)$ represents the class $L$.

To define a symplectic form on $M$, let ${{\omega}}_{{\mathrm{FS}}}$ denote the Fubini–Study form on ${{{{\mathbb C}}\mathrm{P}}}^2$ such that ${{{{\mathbb C}}\mathrm{P}}}^1$ has area $\pi$. Choose an exact perturbation ${{\omega}}_0$ of ${{\omega}}_{{\mathrm{FS}}}$ such that ${{\varphi}}_i^*{{\omega}}_0$ is standard on $B^4_{{\varepsilon}}$ for $i=1,2$. (Warning: ${{\omega}}_0$ cannot be equal to 