Toric variety

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.