The <Cox ring> has one variable for each ray and is $\mathbb C[x,y,z]$, graded by
$$
\deg x=1,\qquad\deg y=a,\qquad\deg z=b.
$$
The irrelevant locus is the common zero of all three variables. The quasitorus associated with the class group is $\mathbb C^*$ and acts by
$$
t\cdot(x,y,z)=(tx,t^ay,t^bz).
$$
The <Cox construction> therefore identifies the closed points of $\mathbb P(1,a,b)$ with
$$
(\mathbb A^3\setminus\{0\})/\mathbb C^*.
$$

Solution

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

 Solution

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