---
date: 2022-01-24 09:52
modification date: Monday 24th January 2022 14:26:56
title: Normalization
aliases: [normalization, normal, "Serre's criterion"]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [integral closure](Unsorted/integrally%20closed.md)

---

# Normalization

- Normal varieties are regular in codimension 1.
	- For curves, this implies nonsingular.
	- For surfaces, this implies no singular curves, only isolated singular points.

![](attachments/Pasted%20image%2020220928230545.png)

![](attachments/Pasted%20image%2020220727093514.png)

- Idea: for affines, $\spec A$ is normal iff $A$ is an integral domain.
- Geometrically: for varieties over fields, $X$ is normal iff any finite birational morphism is an isomorphism.
-
  
- Definition: a scheme is **normal** iff all of its local rings are integral domains which are [integrally closed](Unsorted/integrally%20closed.md).


- Constructing the normalization: take an affine open cover $\ts{\spec A_i}\covers X$, let $B_i = \intcl(A_i)$ be their [integral closures](Unsorted/integrally%20closed.md), and glue.

- Any [reduced scheme](reduced%20scheme) has a unique normalization $\tilde X\to X$ which is birational. 
	- If $\dim X = 1$ then the normalization $\tilde X$ is a [regular scheme](Unsorted/regular%20scheme.md).
	- If $\dim X = 2$ then $\tilde X$ has only isolated singularities.


![](attachments/Pasted%20image%2020220417010850.png)
![](attachments/Pasted%20image%2020220124095237.png)

- [complete](Unsorted/complete.md) curves correspond to taking not just the prime places from the function field, but all places. Can take the projective closure to obtain a complete curve.

![](attachments/Pasted%20image%2020220711105454.png)

# Serre's Criterion

![](attachments/Pasted%20image%2020220905134546.png)

For curves, this yields a [resolution of singularities](Unsorted/resolution%20of%20singularities.md):
![](attachments/Pasted%20image%2020220905134622.png)

Many surface singularities are normal. For example, every hypersurface singularity is $S_2$, so that a hypersurface singularity is normal if and only if it is smooth in codimension one. Similarly, every quotient singularity is normal.

# Examples

- The simplest examples of non-normal varieties: must be dimension 1 or more, so take a curve. If degree 1 or 2, it will automatically be nonsingular and smooth implies normal in dimension 1. So take degree 3 or more, e.g. $V(y^2=x^3+x^2)$ or $V(y^2=x^3)$.