---
date: 2022-09-14 13:13
modification date: Wednesday 14th September 2022 13:13:01
title: "normal"
aliases: [normal]
---

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

---

- Tags: 
	- #AG/basics 
- Refs:
	- #todo/add-references
- Links:
	- [normalization](Unsorted/normalization.md)
	- [DVR](Unsorted/DVR.md)

---

# normal

> But, what is a principal divisor? Here it starts to become clear why one usually assumes that XX is normal even to just talk about divisors, let alone defining the canonical divisor. In order to define principal divisors, one would need to define something like the _order of vanishing_ of a regular function along a prime divisor. It's not obvious how to define this unless the local ring of the general point of any prime divisor is a DVR. Well, then this leads to one to want to assume that $X$ is $R_1$, that is, **regular in codimension 1** which is equivalent to those local rings being DVRs.
> 
> OK, now once we have this we might also want another property: If $f$ is a regular function, we would expect, that the zero set of $f$ should be 1-codimensional in $X$. In other words, we would expect that if $Z \subset X$ is a closed subset of codimension at least 2 , then if $f$ is nowhere zero on $X \backslash Z$, then it is nowhere zero on $X$. In (yet) other words, if $1 / f$ is a regular function on $X \backslash Z$, then we expect that it is a regular function on $X$. This in the language of sheaves means that we expect that the push-forward of $\mathscr{O}_{X \backslash Z}$ to $X$ is isomorphic to $\mathscr{O}_X$. Now this is essentially equivalent to $X$ being $S_2$.
> 
> So we get that in order to define divisors as we are used to them, we would need that $X$ be $R_1$ and $S_2$, that is, **normal**.

![](attachments/Pasted%20image%2020220914131303.png)
![](attachments/Pasted%20image%2020220914131313.png)


![](attachments/Pasted%20image%2020220914182713.png)
![](attachments/Pasted%20image%2020220914182824.png)
![](attachments/Pasted%20image%2020220914182851.png)

# Examples

Should be able to work out explicitly for a nodal or cuspidal curve.