= Solution
The <product fan> for $\mathbb P^1\times\mathbb A^1$ has rays generated by
$$
e_1,qquad e_2,qquad-e_1
$$
and maximal cones $\operatorname{Cone}(e_1,e_2)$ and $\operatorname{Cone}(e_2,-e_1)$. The given fan is obtained by the <star subdivision> of the first maximal cone through
$$
e_1+e_2.
$$
By the <toric blowup at a torus-fixed point>, this subdivision blows up the torus-fixed point corresponding to $\operatorname{Cone}(e_1,e_2)$. Hence
$$
X_\Sigma\cong\operatorname{Bl}_{p}(\mathbb P^1\times\mathbb A^1)
$$
for that torus-fixed point $p$.
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