we have $a_{m}^{\ell}\neq 0$ and $$\hat{v}_{\ell} =
  (a_{0}^{\ell}\,;a_{1}^{\ell}, a_{2}^{\ell},\ldots, a_{m}^{\ell},0,\ldots,0)$$ Because $\hat{v}_{\ell}\cdot\hat{v}_{\ell}\geq 0$ and $d_{\ell}>0$, we must have $m\geq 2$. If $m=2$, then $a_{3}^{N}=0$, so that $a_{3}^{\ell+1}=a_{2}^{\ell}-d_{\ell} = -d_{\ell}<0$, which contradicts the fact that $a_{i}^{\ell}\geq 0$, for all $\ell\geq
0$. Hence, $m\geq 3$. The vectors $(a_{0}^{\ell}\,;a_{1}^{\ell},\ldots,a_{m}^{\ell})$, $\ell>N$, have strictly positive coefficients and thus represent classes in $H_{2}(X_{m};{\mathbb{Z}})$ which, by Corollary \[cor:FundamentalDomain\], must be in $\mathcal{C}_{K}(X_{m})$. By the previous argument, the sequence $\hat{v}_{\ell}$ must be finite.

Strategy for the computation of ${\mathsf{w}}_{k}(M_{\mu}^{i})$ and $p_{k}(M_{\mu}^{i})$ {#sec:strategy}
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Recall that any rational ruled $4$-manifold is, after rescaling, symplectomorphic to either

-   the trivial bundle $M_{\mu}^{0} := (S^2\times S^2,
        {\omega}^0_{\mu})$, where the symplectic area of the a section $S^2\times\{*\}$ is $\mu\geq 1$ and the area of a fiber $\{*\}\times S^{2}$ is 1; or

-   the nontrivial bundle $M_{\mu}^{1} := (S^{2}\ltimes
        S^{2},{\omega}^{1}_{\mu})$, where the symplectic area of a section of self-intersection $-1$ is $\mu>0$ and the area of a fiber is $1$.

We identify the $k$-fold blow-up of $M^0_{\mu}$ of equal sizes $c=c_{1}=\cdots = c_{k}$, with ${\mathbb{CP}}^{2}$ blown up $(k+1)$ times endowed with a symplectic form ${\omega}_{\mu,c}^{0}$ which gives area $\mu+1-c$ to a line and areas $\{\mu-c, 1-c, c,\ldots, c\}$ to the $(k+1)$ exceptional divisors. 