---
date: 2022-02-23 18:45
modification date: Friday 25th February 2022 12:15:15
title: geometric fiber
aliases: [geometric point]

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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

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# geometric fiber

Idea: a point of $X$ with values in an algebraically closed field.

![](attachments/Pasted%20image%2020220423001411.png)

# Geometric Fiber

For $X\in \Sch\slice k$ , a geometric point is a morphism $x: \pt \to X$, where $\pt \da \spec k$ .
The geometric fiber of $x$ is the pullback $E_x \da \lim(\pt \to X \from E) = E\fiberprod{X} \pt$.

Idea: for any $U \subseteq Y$ and $f:X\to Y$, the fiber product $U \fiberprod{Y} X$ should be thought of as $f\inv(U)$.

Note that if $x$ is a finite [etale cover](etale%20cover) , then $\size E_x < \infty$.

![](attachments/Pasted%20image%2020220204100931.png)