---
date: 2022-06-20 15:18
modification date: Monday 20th June 2022 15:18:07
title: "examples of toric varieties"
aliases: [examples of toric varieties]
---

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- Tags: 
	- #research/dynkin-project #AG/toric 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# examples of toric varieties

# Zero: $\GG_m^n$

![](attachments/Pasted%20image%2020220625180536.png)

# A full basis: $\AA^n$

![](attachments/Pasted%20image%2020220625180623.png)



# $\PP^1$

![](attachments/Pasted%20image%2020220625180722.png)

![](attachments/Pasted%20image%2020220620174930.png)

# $\PP^2$

![](attachments/Pasted%20image%2020220625182116.png)
![](attachments/Pasted%20image%2020220625182132.png)

# $\PP^n$

![](attachments/Pasted%20image%2020220620160256.png)
![](attachments/Pasted%20image%2020220620174012.png)
![](attachments/Pasted%20image%2020220620174042.png)
![](attachments/Pasted%20image%2020220620154615.png)




# $\PP^1\times \PP^1$

![](attachments/Pasted%20image%2020220620151809.png)

![](attachments/Pasted%20image%2020220620152526.png)

# Weighted projective spaces 

## $\PP(1,1,2)$

![](attachments/Pasted%20image%2020220620173926.png)

# Cuspidal curves

The cuspidal curve $V\left(x^{3}-y^{2}\right) \cong\left\{\left(t^{2}, t^{3}\right) \mid t \in \mathbb{C}^{2} \subseteq \mathbb{A}^{2}\right.$ is an affine toric variety:
$$
\mathbb{C}^{*}=\left\{\left(t^{2}, t^{3}\right) \mid t \neq 0\right\} \subseteq V\left(x^{3}-y^{2}\right)
$$

# Rational normal curves $C_d$
![](attachments/Pasted%20image%2020220620152423.png)

# Rational normal scrolls
![](attachments/Pasted%20image%2020220620174119.png)

# The Hirzebruch surface $H_r$

![](attachments/Pasted%20image%2020220625182506.png)
![](attachments/Pasted%20image%2020220625182517.png)


# Dual cones and faces

![](attachments/Pasted%20image%2020220620180817.png)

# Cone over a quadric

![](attachments/Pasted%20image%2020220625180654.png)
![](attachments/Pasted%20image%2020220625184149.png)

# Cone over the rational normal curve

![](attachments/Pasted%20image%2020220625194901.png)

# Misc

![](attachments/Pasted%20image%2020220621115247.png)

# $\spec \CC[x,y,z]/\gens{x^2-yz}$
![](attachments/Pasted%20image%2020220621235637.png)
This yields a union of cones:
![](attachments/Pasted%20image%2020220621235819.png)


# Blowups

![](attachments/Pasted%20image%2020220625181946.png)
![](attachments/Pasted%20image%2020220622002411.png)

# Examples of cones and fans

![](attachments/Pasted%20image%2020220621214743.png)
![](attachments/Pasted%20image%2020220621214845.png)

![](attachments/Pasted%20image%2020220621213111.png)


# Exercises
![](attachments/Pasted%20image%2020220625182618.png)

![](attachments/Pasted%20image%2020220806151249.png)