$\mathbb{C}\mathbb{P}^{2} \# \bar{\mathbb{C}\mathbb{P}}^{2}$ is the blow-up of $\mathbb{C}\mathbb{P}^{2}$ at a point. Let $A$ be an affine chart of $ \mathbb{C}\mathbb{P}^{2}$ and $L_{\infty} = \mathbb{C}\mathbb{P}^{2} \setminus A$ be the line at infinity.

Let $p$ be the origin in $A$. Consider the set of all (projective) lines through $p \in \mathbb{C}\mathbb{P}^{2}$, then they cover $\mathbb{C}\mathbb{P}^{2}$ and only intersect pair-wise at $p$. Blowing up $p$ "seperates" all of these lines and creates a $\mathbb{C}\mathbb{P}^{1}$-bundle structure. The base can also be identified with $\mathbb{C}\mathbb{P}^{1}$ since each of the lines containing $p$ intersects the line $L_{\infty}$ at a unique point.

$\mathbb{C}\mathbb{P}^{2} \# \bar{\mathbb{C}\mathbb{P}}^{2}$ is the blow-up of $\mathbb{C}\mathbb{P}^{2}$ at a point. Let $A$ be an affine chart of $ \mathbb{C}\mathbb{P}^{2}$ and $L_{\infty} = \mathbb{C}\mathbb{P}^{2} \setminus A$ be the line at infinity.

Let $p$ be the origin in $A$. Consider the set of all (projective) lines through $p \in \mathbb{C}\mathbb{P}^{2}$, then they cover $\mathbb{C}\mathbb{P}^{2}$ and only intersect pair-wise at $p$. Blowing up $p$ "seperates" all of these lines and creates a $\mathbb{C}\mathbb{P}^{1}$-bundle structure; the fibres being strict transforms of lines through $p$. The base can also be identified with $\mathbb{C}\mathbb{P}^{1}$ since each of the lines containing $p$ intersects the line $L_{\infty}$ at a unique point.