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

Write the invariant prime divisors in the order of the raysas $D_1,\dots ,D_4$. The toric divisor class sequence isand the two characters in the standard basis of $M$ giveThuswhere $S=D_2$, $F=D_1=D_3$, and $D_4=S+kF$. Since the fan is smooth, this is also the Picard group.

The orbit-cone correspondence decomposes ${\mathbb{F}}_k$ into one two-dimensional torus, four one-dimensional torus orbits, and four fixed points. Only the zero-dimensional orbits contribute to the compactly supported Euler characteristic, so

Solved by gpt-5.6-sol high.

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

Solved by gpt-5.6-sol high.

Take the invariant divisor $D_1$, whose class is $F$. Its global sections are represented in the Cox ring by $x_1$ and $x_3$. They have no common zero on $U$, so $F$ is basepoint-free. Its Kodaira map isThis is exactly the ruling ${\mathbb{F}}_k={\mathbb{P}}_{{\mathbb{P}}^1}(\mathcal{O}\oplus \mathcal{O}(k))\to {\mathbb{P}}^1$, and its fibers have divisor class $F$.

Solved by gpt-5.6-sol high.