Let $X\rightarrow \mathbb{P}^2$ be a smooth conic bundle with a non reduced fiber $F$, and $\widetilde{X}$ the blow-up of $X$ along $F$ with exceptional divisor $F\times\mathbb{P}^1$.
I expect $\widetilde{X}$ to be singular along a curve $C\subset F\times\mathbb{P}^1$ mapping two to one onto $\mathbb{P}^1$.
On the other hand I have computed the blow-up of $\mathbb{A}^3$ along the double line $(x^2,y)$ (which should be a local model for the previous situation) and I got that it is singular just along a rational curve mapping isomorphically onto $\mathbb{P}^1$.
What I am doing wrong? Is the blow-up of $\mathbb{A}^3$ along the double line $(x^2,y)$ a local model just for a particular type of conic bundle over $\mathbb{P}^2$ with a non reduced fiber?
Thank you.
