---
date: 2022-05-25 20:04
modification date: Wednesday 25th May 2022 20:04:20
title: "slc"
aliases: [slc, semi log canonical, slc pair]
---

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

---

- Tags: 
	- #todo/untagged
- Refs:
	- Fundamental theorems for slc pairs: 
	  <https://www.math.kyoto-u.ac.jp/~fujino/funda-slc-ag-final.pdf#page=1> #resources/notes 
- Links:
	- #todo/create-links 

---

# slc

A pair $(X, B)$ with $X$ a [reduced](Unsorted/reduced.md) variety and $B = \sum b_i B_i$ a $\QQ\dash$divisor is an **slc pair** iff

- $X$ is $S_2$ (?)
- $X$ has at worst [[double crossings]] in codimension 1,
- the pair $(X^\nu, B^\nu)$ is [[log canonical]] where $\nu: X^\nu \to X$ is the [normalization](Unsorted/normalization.md) and $X^\nu, B^\nu$ are defined by 
  $$\nu^*(K_X + B) = K_{X^\nu} + B^\nu$$
Note that one can write $B^\nu = D + \sum B_i \nu\inv(B_i)$ where $D$ is the [[double locus]].

![](attachments/Pasted%20image%2020220528012726.png)