---
date: 2022-01-15 21:49
modification date: Saturday 22nd January 2022 23:44:35
title: Riemann-Roch
aliases: [Riemann-Roch, complete linear system, "RR"]
---


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

---

- Tags: 
	- #AG 
- Refs:
	- <https://math.mit.edu/~mckernan/Teaching/11-12/Spring/18.726/l_18.pdf#page=2&zoom=auto,-200,797>
- Links:
	- [Serre duality](Unsorted/Serre%20duality.md)

---

# Riemann-Roch

![](attachments/Pasted%20image%2020221123161921.png)

![](attachments/Pasted%20image%2020221123164047.png)
![](attachments/Pasted%20image%2020221123164108.png)

![](attachments/Pasted%20image%2020221111142226.png)

![](attachments/Pasted%20image%2020220913162339.png)
![](attachments/Pasted%20image%2020221106010001.png)


Defining the Chern character:
![](attachments/Pasted%20image%2020220913162407.png)

See [[Chern class]].

## For Curves

![](attachments/Pasted%20image%2020220905135321.png)

## For surfaces

![](attachments/Pasted%20image%2020220905135450.png)


## General

![](attachments/Pasted%20image%2020210510005421.png)

![](attachments/Pasted%20image%2020210510005352.png)

![](attachments/Pasted%20image%2020210626210143.png)

![](attachments/Pasted%20image%2020220214094440.png)
![](attachments/Pasted%20image%2020220214094604.png)
![](attachments/Pasted%20image%2020220214094634.png)
![](attachments/Pasted%20image%2020220214094656.png)


## Examples

![](attachments/Pasted%20image%2020221123165257.png)