Torus-fixed point

A torus-fixed point of a toric variety is a point left unchanged by every element of its acting algebraic torus. In a normal split toric variety $X_{\Sigma }$ of dimension $n$, the orbit-cone correspondence givesConsequently fixed points correspond exactly to full-dimensional cones. Maximality under containment is not enough if a maximal cone has smaller dimension.

For ${\mathbb{A}}^2$ with coordinatewise scaling, the origin is fixed. Points on a punctured coordinate axis have one-dimensional orbits, while points with both coordinates nonzero lie in the dense two-dimensional orbit. For the positive-dimensional torus itself, multiplication has no fixed point.

The fan consisting only of the two positive coordinate rays and zero describes ${\mathbb{A}}^2\setminus \{0\}$. Both rays are maximal cones, but neither is full-dimensional, so there are no fixed points. This example isolates the dimension condition from the informal idea of a maximal face.^[ref]