Suppose $X$ is a smooth projective surface with a dominant morphism $\pi:X \rightarrow \mathbb{P}^{1}$ over a field $k$, where all the fibres of $\pi$ are conics (i.e. a conic bundle). If $\pi$ admits a section $s$ over $k$ (i.e. there exists $s:\mathbb{P}^{1}\_{k}\rightarrow X$ such that $\pi \circ s=\text{Id}\_{\mathbb{P}^{1}}$) then why can't $s$ meet a singular fibre at it's singular point (the singular fibres are two traversal lines, where the intersection of these two lines is the singular point).


Additional question: Why does the singular point on one of the fibres of $\pi$ not define a singular point on $X$. My (very non-rigourus) guess was that as the singular point on this fibre would be 2 dimensional it defines something singular on a conic (which is 1 dimensional) but not on the surface (which is 2 dimensional).



Section of conic bundle
The differential of a section is right inverse to the differential of $\pi$, hence $d\pi$ is surjective and $\pi$ is smooth along the section.


