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Letand let $\tilde{\Sigma }$ be the star subdivision of $\sigma$ through the ray ${\mathbb{R}}_{\ge 0}v$. Its maximal cones areSince $\hat{\phi }(v)=0$, their images lie respectively inThus $\hat{\phi }$ is a morphism of fans from $\tilde{\Sigma }$ to $\Sigma$.

The toric blowup at a torus-fixed point identifiesMore explicitly, the blowup of affine space at the origin is the total space of ${\mathcal{O}}_{{\mathbb{P}}^2}(-1)$. The induced toric morphismis its bundle projection: it sends a nonzero vector to its direction $[x_1:x_2:x_3]$ and restricts to the identity on the exceptional divisor ${\mathbb{P}}^2$. Therefore $g$ resolves the indeterminacy locus of $f$.

A star subdivision inserts the toric ray through a nonzero toric lattice vector $v$ lying in the support of a toric fan. For every toric cone containing $v$, replace it by the toric cones generated by $v$ together with its faces that do not contain $v$, retaining all resulting faces and all unaffected toric cones. The support is unchanged, and the induced toric morphism is proper and birational.

For the first quadrant in ${\mathbb{Z}}^2$, inserting $(1,1)$ splits it into the toric cones generated by $(1,0),(1,1)$ and by $(1,1),(0,1)$. Both determinants are one. This is the toric fan of the blowup of affine space at the origin for ${\mathbb{A}}^2$, with a new invariant exceptional projective line. General star subdivisions need not be blowups of smooth centers; the toric blowup along an orbit closure states the special smooth ray-sum case.^[ref]