SCHEME-THEORIC FIBER

Question

I am reading INTERSECTION THEORY (Fulton), I am in page 15 and I do not understand the meaning of : SCHEME-THEORIC FIBER (I know what is a fiber, but what does scheme-theoric means?

Answer 1

In algebraic geometry, the scheme-theoretic fiber of a morphism of schemes $f: X \to Y$ at $y$ is the fiber product $\kappa(y) \times_Y X$ where $\kappa(y)$ is the residue field at $y$. This is a fiber product in the category of scheme and is a scheme whose underlying topological space is homeomorphic to $f^{-1}(y)$.