---
date: 2022-02-12 02:25
modification date: Saturday 12th February 2022 02:25:54
title: toric
aliases:
  - toric
  - toric variety
  - toric geometry
  - fan
  - polyhedral cone
  - toric varieties
created: 2022-04-05T23:42
updated: 2024-01-16T14:43
---

Last modified: `=this.file.mday`

---

- Tags: 
	- #AG/toric
- Refs:
	- [ ] [Course notes](toric%20stuff%20and%20seems%20ok.pdf)
	- [ ] Introduction to toric varieties:
	      <https://www.youtube.com/watch?v=1w7u5h33_0E> #resources/videos 
	- [ ] Toric varieties lecture:
	      <https://www.youtube.com/watch?v=39YslqpVNLs&list=PLRy_Pn1LtSpfYE39TuCZrJAi7-0z44sQB&index=11> #resources/videos 
	- [ ] Elementary intrroduction: Fulton's *Toric Varieties*  
	      #resources/books #resources/recommendations 
		- Recommended by Valery Alexeev
	- [ ] More advanced: Oda's *Convex bodies in algebraic geometry* #resources/books #resources/recommendations 
		- Recommended by Valery Alexeev
	- [ ] ?
	  <https://arxiv.org/pdf/2012.08666.pdf#page=3> #resources/papers 
	- [ ] Lecture recordings: 
	  <https://www.theorie.physik.uni-muenchen.de/activities/special_lecture_s/toric_geometry_braun/index.html> #resources/full-courses #resources/videos 
	- [ ] MPIM 2018 course in nonlinear algebra:
	      <https://www.youtube.com/watch?v=1EryuvBLY80&list=PLRy_Pn1LtSpejKLClqbrWWBxGKxx333O1> #resources/videos #resources/full-courses 
	- [ ] Talk on toric varieties <https://www.youtube.com/watch?v=bQx4ohtn1oQ>
	- [ ] Borcherds toric varieties <https://www.youtube.com/watch?v=Sjp-99Xyiic>
	- [ ] Introduction to toric geometry with a view towards lattice polytopes Johannes Hofscheier

- Links:
	- [complete intersection](complete%20intersection)
	- [moment map](Unsorted/moment%20map.md)

---

# Toric


![](attachments/Pasted%20image%2020221119214420.png)
![](attachments/Pasted%20image%2020220825111408.png)

# motivation

![](attachments/Pasted%20image%2020220824231429.png)

# toric varieties

Ideas: 
	- A variety $X$ containing a dense open algebraic torus $T$ such that $T\actson T$ extends to an action $T\actson X$.
	- A normal variety $X$ containing a torus $T$ such that $X/T$ has finitely many orbits.
	- A variety described as the closure of the image of a monomial map 
	  $f: (k\units)^m \to \AA^n\slice k$ where $\vector x \mapsto \tv{\vector x^{\vector a_1}, \cdots, \vector x^{\vector a_n}}$.

![](attachments/Pasted%20image%2020220604164331.png)

# Notation

- $M$ is a lattice over a field $k$ with a valuation.
- $N = M\dual \da \Hom_{\zmod}(N, \ZZ)$ is the dual lattice.
- $T_N = N \tensor_\ZZ k\units = \Hom(M, k\units) = (k\units)^n$ is the torus. 
- $N_\RR \da N\tensor_\ZZ \RR \cong \RR^n$ is the real form of $N$.
- $\Sigma$ is a rational fan in $N_\RR$
- $X_\Sigma$ is the associated toric variety completing $T_N$. 
- $\rho_i$ are the rays generating $\Sigma$
- $v_i$ are the first lattice points along the rays $\rho_i$
- $V$ is the matrix whose colums are the $v_i$.
- $\sigma$ are cones in $\Sigma$.
- $S$ is the Cox ring.
- $B$ is the irrelevant ideal.
 
# Fans

- Toric varieties are entirely determined by their associated [[fan]]: a collection of cones closed under taking intersections and faces. In special cases, this is further determined by a polytope.

![](attachments/Pasted%20image%2020220601225425.png)
![](attachments/Pasted%20image%2020220604160831.png)

![](attachments/Pasted%20image%2020220604194928.png)


- A **fan** $\mathcal{G}$ is a family of nonempty closed polyhedral (convex) cones in $V$ such that
	- Every face of a cone in $\mathcal{G}$ is in $\mathcal{G}$, and
	- The intersection of any two cones in $\mathcal{G}$ is a face of both.

Properties:

- A fan $\mathcal{G}$ is **complete** if the union of all its cones is $V$, 
- $\mcg$ is **essential** (or pointed) if the intersection of all non-empty cones of $\mathcal{G}$ is the origin
- $\mcg$ is **simplicial** if every cone is simplicial, that is, spanned by linearly independent vectors. 
- A 1-dimensional cone is called a **ray**.
- A ray is **extremal** if it is a face of some cone. 
- The set of $k$-dimensional cones of $\mathcal{G}$ is denoted by $\mathcal{G}^{(k)}$ and two cones in $\mathcal{G}^{(k)}$ are **adjacent** if they have a common face in $\mathcal{G}^{(k-1)}$. 
- A fan $\mathcal{G}$ **coarsens** a fan $\mathcal{G}^{\prime}$ if every cone of $\mathcal{G}$ is the union of cones of $\mathcal{G}^{\prime}$ and $\bigcup_{C \in \mathcal{G}} C=\bigcup_{C \in \mathcal{G}^{\prime}} C$.


# Expression as a quotient

See <http://jesusmartinezgarcia.net/wp-content/uploads/2019/08/toricNutshell.pdf#page=27>.

![](attachments/Pasted%20image%2020220605122552.png)

# Results

- Every toric variety is rational
	- So a [K3 surface](K3%20surfaces.md) is not toric, since it is not rational.
- ![](attachments/Pasted%20image%2020220604164246.png)

# Limits and completeness

![](attachments/Pasted%20image%2020220824232312.png)

# Examples

- $\AA^n$
- $\PP^n$
- $\prod_i \PP^{n_i}$
- $E\to \PP^n$ a bundle

In [orbifold](Unsorted/orbifold.md) theory, used to define [Hamiltonian](Unsorted/Hamiltonian.md) toric actions:

![](attachments/Pasted%20image%2020220213160506.png)

![](attachments/Pasted%20image%2020220719183840.png)

![](attachments/Pasted%20image%2020220719183503.png)