 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 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 ${\mathbb{A}}^1\setminus \{0\}$.