Torus-fixed point
= 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> gives
$$
\dim O(\sigma)=n-\dim\sigma.
$$
Consequently 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.<Toric Varieties by David A. Cox and others>[Section3.2, orbit-cone correspondence]{ref}