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The Cox ring has one variable for each ray and is $\mathbb{C}[x,y,z]$, graded byThe irrelevant locus is the common zero of all three variables. The quasitorus associated with the class group is ${\mathbb{C}}^{\ast }$ and acts byThe Cox construction therefore identifies the closed points of $\mathbb{P}(1,a,b)$ with

The Cox construction uses coordinates $x_1,\dots ,x_4$ of degreesIts irrelevant locus isso put $U={\mathbb{A}}^4\setminus (V(x_1,x_3)\cup V(x_2,x_4))$. The algebraic torus $G=({\mathbb{G}}_m{)}^2$ acts byEvery point of $U$ has a nonzero coordinate from each opposing pair, so the action has the expected closed orbits and trivial stabilizers. The geometric quotient is

In the Cox ring $S=k[z_{\rho }]$, the irrelevant ideal is generated by ${\prod }_{\rho \notin \sigma (1)}{z}_{\rho }$ as $\sigma$ runs over maximal cones. The Cox construction takes the quotient of $\mathrm{Spec}S\setminus V(B_{\Sigma })$ by the diagonalizable group whose character group is the divisor class group.

// Target: geometry-and-topology.bigb