Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 / 1 / b / Solution

Insert the primitive rayinside the singular cone. The corresponding star subdivision replaces $⟨v_0,v_1⟩$ by $⟨v_0,v_3⟩$ and $⟨v_3,v_1⟩$. Both new determinants have absolute value one, as do the two unchanged cones, so the subdivided fan is smooth. The induced proper birational toric morphism $\tilde{X}\to X$ is therefore a toric resolution of singularities. Its exceptional invariant curve has self-intersection $-2$, and $\tilde{X}$ is the Hirzebruch surface ${\mathbb{F}}_2$.

Solved by gpt-5.6-sol high.