---
created: 2022-04-05T23:42
updated: 2024-01-25T21:07
aliases:
  - geometric genus
  - irregularity
---

- Tags 
	- #AG 
- Refs:
	- Vector bundles on Complex Projective Spaces, by Okonek Schneider Spindler
		- [ ] McKernan's course on this book #resources/full-courses 
			- [ ] [Lecture 1](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_1.pdf)  
			- [ ] [Lecture 2](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_2.pdf)  
			- [ ] [Lecture 3](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_3.pdf)  
			- [ ] [Lecture 4](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_4.pdf)  
			- [ ] [Lecture 5](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_5.pdf)  
			- [ ] [Lecture 6](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_6.pdf)  
			- [ ] [Lecture 7](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_7.pdf)  
			- [ ] [Lecture 8](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_8.pdf)  
			- [ ] [Lecture 9](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_9.pdf)  
			- [ ] [Lecture 10](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_10.pdf)  
			- [ ] [Lecture 11](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_11.pdf)  
			- [ ] [Lecture 12](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_12.pdf)  
			- [ ] [Lecture 13](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_13.pdf)  
			- [ ] [Lecture 14](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_14.pdf)  
			- [ ] [Lecture 15](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_15.pdf)  
			- [ ] [Lecture 16](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_16.pdf)  
			- [ ] [Lecture 17](https://mathweb.ucsd.edu/~jmckerna/Teaching/17-18/Autumn/206A/l_17.pdf)
	- [ ] Lectures on birational geometry:
	  <https://www.dpmms.cam.ac.uk/~cb496/birgeom-paris-public.pdf#page=3> #resources/notes 
	- [ ] Slides from McKernan: 
	  <https://mathweb.ucsd.edu/~jmckerna/Talks/mmp.pdf#page=1> #resources/talks 
	- [ ] Some video lectures #resources/videos 
		- [ ] [Lecture 1](http://www.math.ucsd.edu/~jmckerna/Teaching/20-21/Spring/206A/l_1.pdf)  
		- [ ] [Lecture 2](http://www.math.ucsd.edu/~jmckerna/Teaching/20-21/Spring/206A/l_2.pdf)  
		- [ ] [Lecture 3](http://www.math.ucsd.edu/~jmckerna/Teaching/20-21/Spring/206A/l_3.pdf)  
		- [ ] [Lecture 4](http://www.math.ucsd.edu/~jmckerna/Teaching/20-21/Spring/206A/l_4.pdf)  
		- [ ] [Lecture 5](http://www.math.ucsd.edu/~jmckerna/Teaching/20-21/Spring/206A/l_5.pdf)  
		- [ ] [Lecture 6](http://www.math.ucsd.edu/~jmckerna/Teaching/20-21/Spring/206A/l_6.pdf)  
		- [ ] [Lecture 7](http://www.math.ucsd.edu/~jmckerna/Teaching/20-21/Spring/206A/l_7.pdf)
	- [ ] Notes on cohomology: 
	      <https://people.math.harvard.edu/~mpopa/571/> #resources/notes 
	- [ ] Kodaira dimension:
	      <https://people.math.harvard.edu/~mpopa/483-3/notes.pdf> #resources/notes 
	- [ ] McKernan's UCSD lectures, [[Mori's program]] and higher dimensional geometry #resources/full-courses 
		- [ ] [Lecture 1](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_1.pdf)  
		- [ ] [Lecture 2](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_2.pdf)  
		- [ ] [Lecture 3](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_3.pdf)  
		- [ ] [Lecture 4](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_4.pdf)  
		- [ ] [Lecture 5](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_5.pdf)  
		- [ ] [Lecture 6](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_6.pdf)  
		- [ ] [Lecture 7](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_7.pdf)  
		- [ ] [Lecture 8](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_8.pdf)  
		- [ ] [Lecture 9](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_9.pdf)  
		- [ ] [Lecture 10](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_10.pdf)  
		- [ ] [Lecture 11](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_11.pdf)  
		- [ ] [Lecture 12](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_12.pdf)  
		- [ ] [Lecture 13](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_13.pdf)  
		- [ ] [Lecture 14](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_14.pdf)  
		- [ ] [Lecture 15](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_15.pdf)  
		- [ ] [Lecture 16](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_16.pdf)  
		- [ ] [Lecture 17](https://mathweb.ucsd.edu/~jmckerna/Teaching/14-15/Spring/207A/l_17.pdf)
- Links:
	- [001 Resources References in Algebraic Geometry](Projects/2022%20Algebraic%20Geometry%20Oral%20Exam/001%20Resources%20References%20(AG).md)
	- [[complex algebraic geometry]]
	- [blowup](Unsorted/blowup.md)
	- [MMP](Unsorted/minimal%20model%20program.md)
	- [minimal model](Unsorted/minimal%20model%20program.md)
	- [[Mori fiber space]]
	- [klt](Unsorted/klt.md)
	- [[extremal contraction]]
	- [flips](flips)

---

# birational geometry

Idea: a birational morphism between [schemes](Unsorted/scheme.md) is a morphism that becomes an isomorphism after restricted to some open dense subset.



![](attachments/Pasted%20image%2020220323171822.png)
![](attachments/Pasted%20image%2020220323172025.png)

# Invariants

![](attachments/Pasted%20image%2020220601153518.png)