Let $f: X \to Y$ a smooth, dominant map between two irreducible smooth schemes $X$ and $Y$. let $\eta$ be the generic point of $Y$.
Question: why the codimension of generic fiber $X_{\eta}= f^{-1}(\eta)$ in $X$ coinsides with the dimension of $Y$?
Recall: $codim(X_{\eta},X)= codim(\overline{X_{\eta}},X)$, since in general $X_{\eta}$ is not closed. by definition of codimension, $codim(\overline{X_{\eta}},X)= \max_i(X_i,X)$ where the $X_i$ are irreducible components of $\overline{X_{\eta}}$. therefore, assume that the generic fiber $X_{\eta}$ is irreducible. Can someone please help me how to proceed?