---
date: 2022-04-06 04:42
modification date: Thursday 12th May 2022 21:53:21
title: "Enriques-Kodaira Classification"
aliases: [Enriques-Kodaira classification, Kodaira dimension, Kodaira classification]
---

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

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Kodaira dimension](Unsorted/birational%20geometry.md)

---

# Enriques-Kodaira Classification

- Work over $\CC$ for simplicity, take all dimensions over $\CC$.

- [Minimal model program](Minimal%20model%20program.md) : classifying complex projective varieties.

- Stratify the [moduli space](moduli%20space.md) of varieties by $\kk\dash$dimension.

- Dimension 1:
  - Smooth Algebraic curves = compact Riemann surfaces, classifed by genus
  - Roughly known by Riemann: moduli space of smooth projective [curves](curves.md) $\mcm_g$ is a connected open subset of a projective variety of dimension $3g-3$.

- Dimension 2:
  - Smooth [Algebraic Surfaces](Algebraic%20Surfaces) : Hard. 
    See Enriques classification.
    
  - Setting of classical theorem: always 27 lines on a cubic surface! 

  - Example Clebsch surface, satisfies the system
  \[  
  \begin{array}{l}
  x_{0}+x_{1}+x_{2}+x_{3}+x_{4}=0 \\ \\
  x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=0
  \end{array}
  \]



# Kodaira dimension

![](attachments/Pasted%20image%2020220806235626.png)

![](attachments/Pasted%20image%2020220806235648.png)