Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 / 2 / b

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