A toric variety is a normal variety $X$ over $K$ equipped with a dense open embedding $\mathbb{T}\hookrightarrow X$ and an action $\mu\colon\mathbb{T}\times X\to X$ that extends the action of $\mathbb{T}$ on itself by translations. When we want to stress the torus, we will call $X$ a toric variety with torus $\mathbb{T}$.