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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 $p$.

The inverse image of the dense torus ${\mathbb{A}}^1\setminus \{0\}={\mathbb{G}}_m$ is represented by the subfan formed from cones mapping to the zero cone of the target fan. These areThis is the fan of the projective line in the $e_1$-axis, together with the zero fan in the complementary $e_2$-direction. By the product fan construction,It also follows geometrically because the blowup center lies over zero, so the blowup is unchanged over ${\mathbb{A}}^1\setminus \{0\}$.