Definition
Let \(X\) be a separated scheme of finite type over a field \(k\) and let \(T\) be an algebraic torus over \(k\). We say that \(X\) is a toric variety if
- \(T\subseteq X\) embeds as a dense, open subset and
- the \(T\) action on itself (via multiplication) extends to an action \(T\acts X\).
Some authors require \(X\) to be normal. If you do this, then you get an equivalence of categories between fans and toric varieties.