Surfaces with rational double points Let $S\rightarrow \mathbb{P}^1$ a surface fibered in conics over a field. Assume that $S$ has a single non reduced fiber $F$ with two points of type $A_1$ on it.
Blowing-up the two points and contracting the strict transform of $F$ we get to a smooth surface $S'\rightarrow\mathbb{P}^1$ which is a conic fibration whose fibers are all reduced.
Now, assume that the surface $S\rightarrow \mathbb{P}^1$ has just one singular point on $F$ but of type $D_n$ ($D_4$ for instance). Can we get a smooth surface $S\rightarrow\mathbb{P}^1$ birational to $S$ over $\mathbb{P}^1$ and whose fibers are all reduced?
 A: Yes, this can be done. For intance, begin with a surface $S$ with a non reduced fiber $F$ and two points of type $A_1$.
Take a smooth point $x\in F$ and let $W$ be the blow-up of $S$ at $x$ with exceptional divisor $E$. Denote by $C$ the strict transform of $F$. Then there is a morphism $W\rightarrow T$ contracting $C$ to a point $z\in T$. If $\widetilde{T}$ is a minimal resolution of $T$ then there is a morphism $\widetilde{T}\rightarrow W$ and hence a morphism $\widetilde{T}\rightarrow S$.
Let $\widetilde{S}\rightarrow S$ be a minimal resolution of $S$. Then $\widetilde{T}$ is the blow-up of the inverse image $x'$ of $x$ in $\widetilde{S}$.
In $\widetilde{S}$ we have a tree of curves $E_1\cup \widetilde{F}\cup E_2$ where $E_1,E_2$ are the exceptional curves and $\widetilde{F}$ is the strict transform of $F$. Blowing-up $x'$ we get a configuration of rational curves in $\widetilde{T}$ given by $\widetilde{E}_1\cup \widetilde{\widetilde{F}}\cup \widetilde{E}_2\cup E_3$ where $\widetilde{\widetilde{F}}$ intersects $\widetilde{E}_1$, $\widetilde{E}_2$ and $E_3$ but these last three curves do not intersect each other. Hence $z\in T$ is a point of type $A_3$ (or $D_3$).
Now, denote by $\Gamma\subset T$ the image of $E$. If you take a smooth point $x\in \Gamma$ blow it up and contract the strict transform of $\Gamma$ you get a surface $T_1$ with a singular point of type $D_4$. Prooceeding in this way you can go from a singularity of type $D_i$ to one of type $D_{i+1}$.
Looking at the construction backwards you can start from a surface $S\rightarrow\mathbb{P}^1$ with a non reduced fiber and a point of type $D_n$ on it and get to a smooth surface $\overline{S}\rightarrow\mathbb{P}^1$ whose fibers are reduced conics with a finite number of birational transformations (over the base $\mathbb{P}^1$) as I described.
