= Solution
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 are
$$
\Sigma_0=\{0,\operatorname{Cone}(e_1),\operatorname{Cone}(-e_1)\}.
$$
This 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,
$$
X_{\Sigma_0}\cong\mathbb P^1\times\mathbb G_m
=\mathbb P^1\times\mathbb C^\times.
$$
It also follows geometrically because the blowup center lies over zero, so the blowup is unchanged over $\mathbb A^1\setminus\{0\}$.
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