The cone $\sigma=\operatorname{Cone}(e_1,e_2,e_3)$ and all its faces form the standard fan of affine three-space, so
$$
U_\sigma\cong\mathbb A^3.
$$
The complete target fan has primitive rays
$$
f_1,qquad f_2,qquad-f_1-f_2
$$
and every cone generated by a proper subset of them. This is the two-dimensional <simplex fan>, hence
$$
X_\Sigma\cong\mathbb P^2.
$$

Solution

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

 Solution

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