Let $C=Spec \mathbb{C}[t]/(t^{n+1})$. Let $X$ be an algebraic (or complex analytic) scheme, flat over $C$ with the structure morphism $f\colon X\to C$. Assume that the special fiber is isomorphic to $\mathbb{C}\mathbb{P}^1$. **Is it true that there exits an isomorphism $g\colon X\tilde\to \mathbb{C}\mathbb{P}^1\times C$ of $C$-schemes?** 


A reference would be very helpful.



Flat family with special fiber $\mathbb{C}\mathbb{P}^1$
Yes. $X$ is a Severi-Brauer scheme over $C$ (see this [Bourbaki talk](http://archive.numdam.org/ARCHIVE/SB/SB_1964-1966__9_/SB_1964-1966__9__199_0/SB_1964-1966__9__199_0.pdf) by Grothendieck, Cor. 8.3). Since any finite étale covering of $C$ is trivial, this implies by definition (loc. cit.) that $X$ is $C$-isomorphic to $C\times \mathbb{P}^1$.


