---
date: 2022-01-28 15:06
modification date: Friday 28th January 2022 22:33:12
title: simple normal crossings
aliases:
  - SNC
  - SNC divisor
  - simple normal crossings divisor
  - simple normal crossings divisor
  - strict normal crossings
  - strict normal crossings divisor
  - normal crossings
  - normal crossings divisor
  - NC
created: 2022-04-05T23:42
updated: 2024-01-31T14:14
---
---
- Tags: 
	- #AG/divisors
- Refs:
	- #todo/add-references
- Links:
	- [resolution of singularities](Unsorted/resolution%20of%20singularities.md)
	- [[log vector field]]
	- [transverse](Unsorted/transverse.md)

---

# simple normal crossings divisor

- NC: $X$ is **normal crossings** at $x$ iff there is an isomorphism on formal neighborhoods $(\hat X, a)\iso (\hat Y, 0)$ to a union of coordinate hyperplanes in $\AA^n$ at zero.
- SNC: $X$ is **simple normal crossings** at $x$ iff it is NC at $x$ and all components $X_i \subseteq X$ passing through $a$ are [[regular]] at $a$.

Warning: an NC scheme need not be [[reduced]].

For [[complex analytic spaces]], a divisor $D$ is **normal crossings** at $p$ iff there are coordinates $\ts{x_i}_{i\leq n}$ such that locally at $p$, $D$ is defined by $V(\prod_{i\leq m} x_i)$ for some $m$.

![](attachments/Pasted%20image%2020220529010622.png)

![](attachments/Pasted%20image%2020220529011304.png)

![](attachments/Pasted%20image%2020220528012830.png)
![](attachments/Pasted%20image%2020220528012903.png)

![](2024-01-31-9.png)
# Results

![](attachments/Pasted%20image%2020220529012224.png)
![](attachments/Pasted%20image%2020220529012543.png)
![](attachments/Pasted%20image%2020220529012627.png)