What are the irreducible curves on the blow up of $\mathbb{P}^{2}$? On the blow-up of $2$-dimensional complex projective space, $\mathbb{P}^2$, I know that
$$Pic(\mathbb{\tilde{P}^2})=\mathbb{Z}[\tilde{H}]+\mathbb{Z}[E]$$
where $\tilde{H}$ is the blow-up of the hyperplane bundle of $\mathbb{P}^2$, and $E$ is the exceptional divisor on $\tilde{\mathbb{P}}^2$. Note that $[\cdot]$ denotes numerical equivalence class. 
I already know that the set $\{[\tilde{H}], [E]\}$ form a basis for $\tilde{\mathbb{P}}^2$. but how do the irreducible curves on the blow-up look like?

They are either strict transforms of irreducible curves from $\mathbb{P}^2$, or they are the exceptional divisor $E$.
