Solution
= Solution
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
$$
(\mathbb A^3\setminus\{0\})/\mathbb C^*.
$$