---
date: 2022-02-23 18:45
modification date: Monday 4th April 2022 00:47:32
title: "Zariski's main theorem"
aliases: [Zariski's main theorem]

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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Artin approximation](Unsorted/Artin%20approximation.md)
	- [Noether normalization](Unsorted/Noether%20normalization.md)
	- [Hilbert Basis Theorem](Unsorted/Hilbert%20Basis%20Theorem.md)
	- [Stein factorization](Unsorted/contraction%20(of%20curves).md)

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# Zariski's main theorem

Theorem 8.6. Let $f: X \rightarrow Y$ be a birational morphism of normal projective varieties over $\mathbb{C}$.

(a) (Zariski Main Theorem) For any point $y \in Y$ the fiber $f^{-1}(y)$ is connected. In particular, $\operatorname{dim} f^{-1}(y)>0$ unless $f^{-1}(y)$ is a singleton.
(b) (Purity Theorem) The exceptional locus of $f$,
$$
E=E(f):=\bigcup_{\operatorname{dim} f^{-1}(y)>0} f^{-1}(y),
$$
is Zariski closed in $X$, and if $y_{0} \in f(E)$ is a smooth point of $Y$, then $E$ has codimension 1 in $X$ near the points of the fiber $f^{-1}\left(y_{0}\right)$. If $Y$ is smooth, then $E$ is a projective hypersurface.

![](attachments/Pasted%20image%2020220601125954.png)
![](attachments/Pasted%20image%2020220404004735.png)
![](attachments/Pasted%20image%2020220407235015.png)

# Applications
![](attachments/Pasted%20image%2020220211121734.png)