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For a 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 $\sigma \in \Sigma$, withThe 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 $(k^{\ast }{)}^2$, 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. ^[ref]

A primitive lattice vector is a nonzero element $u$ of a finite-rank free abelian group $N$ that cannot be written $u=kv$ with $v\in N$ and an integer $|k|>1$. In an integral basis, this is equivalent to the greatest common divisor of its coordinates being one. The property is independent of the chosen lattice basis.

For example, $(2,3)$ is primitive although neither coordinate is one: the integer functional $\ell (x,y)=-x+y$ satisfies $\ell (2,3)=1$. Hence $N=\mathbb{Z}u\oplus \ker \ell$, showing that $u$ extends to a lattice basis. In contrast, $(2,4)=2(1,2)$ is not primitive.

Every rational ray of a fan contains a unique primitive generator pointing along it. This generator gives the integral normalization used in toric divisors and cocharacters; replacing it by a larger positive multiple leaves the geometric ray unchanged but changes its lattice data.

A ray of a fan is a one-dimensional cone in a fan in toric geometry. It has the form ${\mathbb{R}}_{\ge 0}v$, where $v$ is a primitive lattice vector, and occurs as a face of a polyhedral cone.

A ray of a fan is a one-dimensional cone. It has a unique primitive lattice generator $v_{\rho }$ pointing along it.