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created: 2023-06-07T18:18
updated: 2023-06-07T18:19
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date: 2022-02-23 18:45
modification date: Thursday 17th March 2022 21:33:24
title: Log geometry
aliases: [Log geometry]

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- Tags 
	- #todo/untagged
- Refs:
	- Mattia Talpo: Kummer-étale additive invariants of log schemes. <https://www.youtube.com/watch?v=r-CXf24pE9Y>
- Links:
	- #todo/create-links 

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# Log geometry

![](2023-06-07.png)

![](attachments/Pasted%20image%2020220317213329.png)
![](attachments/Pasted%20image%2020220317213350.png)
# Notes

> Reference: Mattia Talpo: Kummer-étale additive invariants of log schemes. <https://www.youtube.com/watch?v=r-CXf24pE9Y>

- Origins: Fontaine-Illusie-Kato (Deligne, Faltings) in the late 80s, schemes/stacks/derived schemes plus additional derived ("log") structure.
- Examples:
	- A pair $(X, D)$ of a [smooth divisor](smooth%20divisor).
	- Or $D$ a toroidal embedding, remembers the boundary $D$
		- [toric](toric.md) boundary
	- Think about this like ""$X$ with a marked point on the boundary"
- How they arise: in characteristic zero, working with a non-compact scheme. Compactify, and use [divisor](divisor.md).
- Every log scheme has a locus where the log structure is trivial, $\ts{x\in X\st \OO_{X, x}\units = M_x}$, so [stalk of a sheaf](stalk%20of%20a%20sheaf) of $M$ are units. Think of this like the complement of the boundary.
	- Trivial part of a log scheme pair $(X, D)$ is the complement of the divisor.
		- Rank of the [free monoid](free%20monoid) records number of branches passing through singular points:

	![](attachments/Pasted%20image%2020210515182026.png)

- Logarithmic geometry generalizes [toroidal](toroidal) geometry to non-smooth settings.
- See [semistable degeneration](semistable%20degeneration.md)