---
tag: E22Y744R1C
kind: definition
name: Toric variety
msc: [14M25]
defines: YWB981MEF2
uses: [66FQFAJNS4, RW85C34GQW, HV6YBS0YDH, EK261G1XP9]
sources:
  - url: https://en.wikipedia.org/wiki/Toric_variety
---

## Definition

Let $k$ be a field, and let $T \cong \mathbb{G}_m^n$ be a split [[algebraic torus|66FQFAJNS4]] over $k$.
A *toric variety* with torus $T$ is a [[variety|HV6YBS0YDH]] $X$ over $k$ with an open embedding $T \hookrightarrow X$ such that:

1. $X$ is [[normal|EK261G1XP9]];
2. the image of $T$ is dense in $X$;
3. the action of $T$ on itself by multiplication extends to an [[action|RW85C34GQW]] $T \times X \to X$.

## Remarks

Some authors drop clause 1 and speak of normal toric varieties when it holds.
Normal toric varieties correspond to fans: finite collections of strongly convex rational polyhedral cones in $\R^n$, closed under faces, any two of which meet in a common face.

Example: $\mathbb{P}^n$ with the torus $T = \set{[x_0 : \cdots : x_n] : x_0 x_1 \cdots x_n \ne 0} \cong \mathbb{G}_m^n$.
Here $T$ is open and dense, $\mathbb{P}^n$ is smooth and hence normal, and $(t_1, \ldots, t_n) \cdot [x_0 : x_1 : \cdots : x_n] = [x_0 : t_1 x_1 : \cdots : t_n x_n]$ extends the multiplication.

Non-example: the cuspidal curve $C = \set{y^2 = x^3}$ in $\mathbb{A}^2$ contains the dense open torus $\set{(t^2, t^3) : t \ne 0}$.
The action $s \cdot (x, y) = (s^2 x, s^3 y)$ extends the multiplication, so clauses 2 and 3 hold.
But $C$ is not normal at the origin, so it fails clause 1.