---
date: 2022-02-11 12:19
modification date: Friday 11th February 2022 12:19:37
title: resolution of singularities
aliases: [resolution of singularities, crepant resolution]

---

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

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [SNC](Unsorted/simple%20normal%20crossings.md)
	- [strict normal crossings divisor](strict%20normal%20crossings%20divisor.md)
	- [alteration](alteration.md)

---

# resolution of singularities

![](attachments/Pasted%20image%2020221110221743.png)
![](attachments/Pasted%20image%2020220913155335.png)

Hironaka: every [projective variety](projective%20variety) is birational to a [smooth](Unsorted/smooth%20scheme.md) projective variety.

![](attachments/Pasted%20image%2020220411231323.png)

![](attachments/Pasted%20image%2020220529010847.png)

![](attachments/Pasted%20image%2020220411231359.png)
![](attachments/Pasted%20image%2020220411231422.png)

# Misc
![](attachments/Pasted%20image%2020220212190249.png)

See [regular scheme](regular%20scheme), [proper](Unsorted/proper%20morphism.md), [birational](Unsorted/birational.md) morphisms.
![](attachments/Pasted%20image%2020220211121940.png)
![](attachments/Pasted%20image%2020220211122344.png)

![](attachments/Pasted%20image%2020220213143036.png)

![](attachments/Pasted%20image%2020220325101403.png)