---
created: 2023-03-28T21:18
updated: 2024-01-01T23:15
title: minimal model program
aliases:
  - minimal model program
  - minimal model
  - MMP
  - birational minimal model
---


---

- Tags 
	- #todo/untagged
- Refs:
	- <https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/> #resources/full-courses 
	- Introduction to MMP by Kawamata et al: <https://projecteuclid.org/ebook/download?urlId=10.2969/aspm/01010283&isFullBook=false#page=1> #resources/notes 
	- 2021 Course notes: <https://www.math.columbia.edu/~plei/docs/MMP.pdf#page=2> #resources/course-notes 
	-  _Positivity in Algebraic Geometry_ I & II, by Lazarsfeld
- Links:
	- [Chevalley-Eilenberg complex](Chevalley-Eilenberg%20complex.md)
	- [Quillen minimal model program](Quillen%20minimal%20model%20program.md)
	- [nef divisor](Unsorted/nef%20divisor.md)
	- [Gorenstein](Unsorted/Gorenstein.md)
	- [[abelian surface]], [[Jacobian]]
	- [[K3 surface]], [canonical divisor](Unsorted/canonical%20divisor.md)
	- [Enriques surface](Enriques%20surface.md)
	- [[bielliptic surface]]
	- [[elliptic surface]]
---

# MMP

![](2023-03-28-13.png)
![](2023-03-28-15.png)

![](attachments/Pasted%20image%2020220531035023.png)
![](attachments/Pasted%20image%2020220531035046.png)

![](attachments/Pasted%20image%2020220528011739.png)




Blowups aren't usually minimal models:

![](attachments/Pasted%20image%2020220531035135.png)

So restrict attention to **minimal** surfaces: those that are not the [[blowup]] of any smooth surface.

![](attachments/Pasted%20image%2020220531035256.png)

See:


# Birational minimal model program

See [flips](flips)
![](attachments/Pasted%20image%2020220323172119.png)

![](attachments/Pasted%20image%2020220422224915.png)



# Results

![](attachments/Pasted%20image%2020220515001512.png)


# The algorithm
![](attachments/Pasted%20image%2020220514203555.png)![](attachments/Pasted%20image%2020220514203653.png)

# Status

![](attachments/Pasted%20image%2020220601125826.png)