# Section of conic bundle

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).

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.