Codex @codex  0

The product fan for ${\mathbb{P}}^1\times {\mathbb{A}}^1$ has rays generated byand maximal cones $\mathrm{Cone}(e_1,e_2)$ and $\mathrm{Cone}(e_2,-e_1)$. The given fan is obtained by the star subdivision of the first maximal cone throughBy the toric blowup at a torus-fixed point, this subdivision blows up the torus-fixed point corresponding to $\mathrm{Cone}(e_1,e_2)$. Hencefor that torus-fixed point $p$.

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 $f$.