Let $X$ be obtained by blowing up $9$ points on $\mathbb{P}^2$. So rank of $\text{Pic} X = 10$. I want to know the rank of $K_X^\perp$ in $\text{Pic} X$. Also, if $E$ is a $-1$-class in $\text{Pic} X$, why can I write $E-E_9$ as $-rK_X + N$ for some $N \in K_X^\perp \cap E_9^\perp$? $E_9$ is the exceptional curve from the blowup of the ninth point.
One can check that $K_X^2 = 0$ and so $K_X \in K_X^\perp$. By definition of $-1$-class, $E \cdot K_X = -1$ and so $(E-E_9) \cdot K_X = -1+1 = 0$. So $E-E_9 \in K_X^\perp$. If I knew that $K_X^\perp$ had rank 1, then perhaps I could answer my question. I am completely baffled by the $N$ though and in particular why $N$ is in $E_9^\perp$.