Generic fiber ($X_{\eta }=X{\times }_S\mathrm{Spec}\kappa (\eta )$)

For a morphism of schemes $X\to S$ with $S$ an integral scheme and generic point $\eta$, the generic fiber is its scheme-theoretic fiber over $\eta$. On affine schemes $S=\mathrm{Spec}A$ and $X=\mathrm{Spec}B$, it is $\mathrm{Spec}(B{\otimes }_A\mathrm{Frac}A)$. It can be empty, and it need not be a reduced scheme or an irreducible scheme without further assumptions.