Let $p:Bl_0(\mathbb{A}^2) \to \mathbb{A}^2$ be blowup of plane at the origin. Is there a geometric reason why there are no sections $s: \mathbb{A^2} \to Bl_0(\mathbb{A^2})$, that is no maps to vareties with $p \circ s= id$ on affine plane $\mathbb{A^2}$.
Where I'm trying to get to is: in this comment by Zhen Lin used this recognizing the total space of the tautological line bundle $O(-1)$ on $\mathbb{P}^1$ to be identical to the blowup of affine plane at the origin.
Here's one simple geometric reason: Consider any two distinct lines $L_1, L_2$ through the origin in $\newcommand{\AA}{\mathbb{A}}\AA^2$. The proper transforms of these two lines under the blowup at the origin are two non-intersecting lines in $\newcommand{\Bl}{\operatorname{Bl}}\Bl_0(\AA^2)$. If there were a section $s\colon \AA^2 \to \Bl_0(\AA^2)$ of the blowup map, then it would have to map each of these lines to its proper transform, because the image of a connected space under a continuous map is connected, and regular maps are continuous. But this is impossible, because this forces $s(0, 0)$ to have two different values.