# Construction of a projective variety with a given generic fiber

Let $C$ be a projective regular integer curve over a (finite) field $k$ and $K:=K(C)$ the function field. Let $f\colon Y\rightarrow C$ be a projective variety with generic fiber $j\colon Y_\eta\rightarrow Y$ and $i\colon X\hookrightarrow Y_\eta$ a closed immersion. Define $X'$ as the zariski-closure of $j(i(X))$ in $Y$ with the reduced structure.

Now I want to show, that $X'\rightarrow C$ is a projective variety with generic fiber $X$.

As $X'\rightarrow Y$ should be a closed immersion it is clear, that $X'\rightarrow C$ is projectiv. Because $X$ is irreducible it follows, that $X'$ is irreducible, too. And because it is reduced, we get integer.

Now to the generic fiber of $X'$. That's the point, I'm no longer sure, if my argumentation holds: Topologically, the fiber $X'\times_C\text{Spec} K$ is $\{x\in X'\mid x\mapsto\eta$ under $X'\rightarrow C\}$, and because $Y_\eta=Y\times_C\text{Spec }K$ the fiber $X'\times_C\text{Spec }K$ should be the smallest closed set in $Y_\eta$, that contains $i(X)$. But $i$ is a closed immersion, and therefore $i(X)=X=X'\times_C\text{Spec }K$ topologically. If I could show that the fiber is reduced, it should follow, that $X\cong X'\times_C\text{Spec }K$. But here's the point I'm stuck.

EDIT Local ring on generic fiber Here is the proof, that the local rings of the fiber are reduced (because $X'$ is reduced). Now the only question is, if my topological argumentation is right.