---
date: 2022-01-15 21:49
modification date: Friday 28th January 2022 18:18:17
title: special fiber
aliases: [special point, closed point, closed fiber]

---

Tags: 
#AG 
Refs:
[degeneration](Unsorted/degeneration.md)



# special fiber

- For $R\in \Loc\Ring$ a [local ring](Unsorted/localization%20of%20rings.md), $\spec R$ has 
	- A **generic point** $\gens 0$ and 
	- A **special point**  (or **closed point**) $\mfm_R$.
- Easiest example: for $R\in \DVR$, note $\spec R = \ts{\gens 0, \mfm_R}$ has a single generic point and a single closed point.
- For $R\in \DVR$ or $\Loc\Ring$ and a morphism $f: X\to \spec R$, 
	- The **generic fiber** is the fiber over the generic point $(0)$, so $f\inv(\gens 0)$ 
	- The **special fiber** is the fiber above $\mfm$, so $f\inv(\mfm)$.
- [degeneration](Unsorted/degeneration.md) is the theory of passing from the generic fiber to the special fiber.

# Closed points

![](attachments/Pasted%20image%2020220214093029.png)

![](attachments/Pasted%20image%2020220424122108.png)