Let $ X\rightarrow Spec(\mathbb{C}[t])=\mathbb{C}$ be a projective variety over $\mathbb{C}[t]$ (a flat family of projective varieties $X\_{t}$, $t\in \mathbb{C}$), and $X\_{\eta}$ be the base change via $\mathbb{C}[t] \rightarrow \mathbb{C}((t))$ by $t\mapsto t$. ($X\_{\eta}$ is the formal neighborhood of $X\_{0}$ in $X$.) Let $ Y\rightarrow Spec(\mathbb{C}[[t]])$ be a projective variety over $\mathbb{C}[[t]]$, $Y\_{0}$ be the special fiber, and $Y\_{\eta}$ be the general fiber, i.e. the base change via $\mathbb{C}[[t]] \rightarrow \mathbb{C}((t))$. Assume that $X\_{\eta}$ is isomorphic to $Y\_{\eta}$ as varieties over $\mathbb{C}((t))$. 


**The question:** Is there a projective variety $Z \rightarrow Spec(\mathbb{C}[t])$ over $\mathbb{C}[t]$ such that $Z\_{t}$ is isomorphic to $X\_{t}$ for any $t\neq 0$, and $Z\_{0}$ is isomorphic to $Y\_{0}$? (A sort of surgery to replace $X\_{0}$ by $Y\_{0}$.)


Thanks in advance! 



Possible `surgery' via formal neighborhoods
That's fpqc-descent. For this to work you need additionally ample line bundles on both $X\_{\ne0}$ and $Y$ and a compatible extension of the isomorphism $X\_\eta\to Y\_\eta$ to these line bundles. See SGA 1,Proposition 7.8.


