Zariski’s Main Theorem

Abstract

Let ϕ : V → W be a morphism of (quasi-projective) varieties. For y ∈ W, the closed subset of V, namely,

$${\varphi ^{ - 1}}\left( y \right) = \left\{ {x \in V\left| {\varphi \left( x \right) = y} \right.} \right\}$$

is called the fibre of ϕ over y. It is a closed algebraic subset of V, equipped with the canonical reduced structure (22.3).