---
date: 2022-04-21 23:13
modification date: Thursday 21st April 2022 23:13:47
title: moduli space
aliases:
  - moduli space
  - fine moduli space
  - coarse moduli space
  - moduli problem
created: 2022-04-21T23:13
updated: 2024-05-15T17:50
---

# moduli space

![](2024-01-10.png)

![](attachments/Pasted%20image%2020220503102938.png)
![](attachments/Pasted%20image%2020220503103316.png)

- One can say a great deal about the moduli space purely in terms of the moduli functor without even knowing the moduli space exists 
	- For example, the [dual numbers](dual numbers). $\spec k\dualnumbers$

- Prototypical example of a moduli space: the [Grassmannian](Grassmannian.md)  $\Gr_k(\CC^n)$.
- Common example: the [Hilbert scheme](Hilbert%20scheme.md).
- Apparently a [fundamental class](fundamental%20class.md) exists for closed subvarieties? 
	Maybe just closed subvarieties of a moduli space?
- Prominent example when studying [elliptic curves](MOCs/elliptic%20curve.md) over $\QQ$: the [modular curve](Unsorted/modular%20curve.md) $X_0(n)$.
	- [level structure](Unsorted/level%20structure.md): examples are $\Gamma(N), \Gamma_1(N), \Gamma_0(n)$ for squarefree $n$.

## Fine moduli spaces

![](2024-01-10-1.png)

The first condition determines $M$ as a topological space (point set); condition
b) determines the scheme structure on $M$ uniquely.

![](attachments/Pasted%20image%2020220806134815.png)


![](attachments/Pasted%20image%2020210627224433.png)

![](attachments/Pasted%20image%2020220322153642.png)

![](attachments/Pasted%20image%2020220322153755.png)

# Examples

![](attachments/Pasted%20image%2020220322153853.png)
![](attachments/Pasted%20image%2020220322153907.png)

# Counterexamples

![](attachments/Pasted%20image%2020220322153829.png)
![](attachments/Pasted%20image%2020220322154015.png)

# Examples

- $\mbar_{0, 4}\cong \PP^1\slice \CC$ using the classical cross-ratio.