The <toric divisor class sequence> is
$$
M\longrightarrow\mathbb ZD_1\oplus\mathbb ZD_2\oplus\mathbb ZD_3
\longrightarrow\operatorname{Cl}(X)\longrightarrow0.
$$
The two characters give
$$
D_1-aD_3=0,
\qquad
D_2-bD_3=0.
$$
Thus
$$
\operatorname{Cl}(\mathbb P(1,a,b))\cong\mathbb Z,
$$
generated by $[D_3]$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 152 / 2 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-152/2/a/ii/solution