Let $X$ be a projective, geometrically connected $k$-surface with a relatively minimal conic bundle structure $X \longrightarrow \mathbb{P}^1_k$. My understanding is that the generic fiber ought to be a smooth, genus zero curve and say we have n closed points of $\mathbb{P}^1_k$ with non-smooth fiber. Over the residue fields at those points, these singular fibers decompose into two lines intersecting transversally at a rational point.
Can someone help me to understand that Picard group Pic$(\overline{X})$ where $\overline{X} = X \times_k \text{Spec } \overline{k}$ for $\overline{k}$ a separable closure of $k$? Especially interested, in how we use the components of these irreducible fibers. Moreover, how do we understand the action of the Galois group $Gal(\overline{k}/k)$ on Pic$(\overline{X})$?
