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 $p$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 152 / 2 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-152/2/a/i/solution