= Orbit-cone correspondence
= Torus orbit-cone correspondence
{synonym}
For a <normal variety>[normal] split <toric variety> $X_\Sigma$ of <dimension> $n$, the <orbit-cone correspondence> assigns an <orbit> $O(\sigma)$ of the <algebraic torus> to each <toric cone> $\sigma\in\Sigma$, with
$$
\dim O(\sigma)=n-\dim\sigma.
$$
The zero <toric cone> gives the dense <algebraic torus>; a <ray of a fan> gives a codimension-one <orbit>. The <torus-fixed points> correspond exactly to full-dimensional <toric cones>. A maximal <toric cone> need not be full-dimensional: the <toric fan> containing only the zero <toric cone> gives the <algebraic torus> itself and has no fixed <point> when $n>0$.
For <affine space> $\mathbb A^2$, the first-quadrant <toric fan> has a two-dimensional zero-cone orbit $(k^*)^2$, two one-dimensional <toric ray> orbits on the punctured axes, and one zero-dimensional orbit at the origin. The face-to-orbit reversal also explains the formula for an <orbit closure in a toric variety>. <Toric Varieties by David A. Cox and others>[section3.2, orbit-cone correspondence]{ref}
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