Fiber of a morphism

Summary

The fiber of a morphism of schemes $f \colon X \to Y$ over a point $y$ is a scheme $X_y$ over the residue field $\kappa(y)$. Its underlying space is the preimage $f^{-1}(y)$, but it also carries a scheme structure, which may be non-reduced. A morphism is often thought of as a family of schemes, its fibers, parametrized by the points of $Y$.