and A.-K. Liu characterized the symplectic cone of smooth, closed, oriented 4-manifolds with $b^{+}=1$ in terms of exceptional classes. In the case of $X_{n}$, this gives

\[thm:CharacterizationEK\] $$\mathcal{C}_{K} =
    \{A\in{\mathcal{P}}_{+}~|~A\cdot E >0\text{~for all~}E\in{\mathcal{E}}_{K}\}.$$

Since ${\mathcal{E}}_{K}$ is not explicitly known for $n\geq 10$, this characterization cannot be used directly to show that a given class $A\in{\mathcal{P}}_{+}$ belongs to $\mathcal{C}_{K}$. However, the group $\operatorname{Diff}_{+}$ of orientation preserving diffeomorphisms acts on $H_{2}(X_{n};{\mathbb{Z}})$, and any diffeomorphism preserving $K$ also preserves the sets ${\mathcal{E}}_{K}$ and $\mathcal{C}_{K}$. Let us write ${\mathcal{O}}(1,n)$ for the group of orthogonal transformations of $H_{2}(X_{n};{\mathbb{Z}})$, $D(1,n)$ for the image of $\operatorname{Diff}_{+}$ in ${\mathcal{O}}(1,n)$, and $D_{K}(1,n)$ for the subgroup of $D(1,n)$ fixing $K$. Recall that if a class $A\in H_{2}(X_{n};{\mathbb{Z}})$ of self-intersection $\pm1$ or $\pm2$ is represented by a smooth embedded sphere, then the reflection about $A$ $$r_{A}(B):=B - 2\left(\frac{A\cdot B}{A\cdot A}\right)A$$ belongs to $D(1,n)$. Assume $n\geq 3$ and set $$\label{def:BasicRoots}
\begin{aligned}
  \alpha_{0}&=L-E_{1}-E_{2}-E_{3}\\
  \alpha_{i}&=E_{i}-E_{i+1}\,, \quad 1\leq i\leq n-1.
\end{aligned}$$ For $n\geq 3$, those classes are represented by smooth embedded spheres, and since $\alpha_{i}\cdot\alpha_{i} = -2$ and $K\cdot\alpha_{i}=0$, the reflections $r_{\alpha_{i}}$ belong to $D_{K}(1,n)$. The reflexion $\mathfrak{C}:=r_{\alpha_{0}}$, classically known as the *Cremona* transformation, takes a class $\left(a_{0}\,; a_{1},\ldots,a_{n}\right)$ to the class $$\left(a_{0}-d\,; a_{1}-d,a_{2}-d, a_{2}-d, 