Insert the primitive ray
$$
v_3=(0,-1)=\frac12(v_0+v_1)
$$
inside the singular cone. The corresponding <star subdivision> replaces $\langle v_0,v_1\rangle$ by $\langle v_0,v_3\rangle$ and $\langle v_3,v_1\rangle$. 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> $\widetilde X\to X$ is therefore a <toric resolution of singularities>. Its exceptional invariant curve has self-intersection $-2$, and $\widetilde X$ is the <Hirzebruch surface> $\mathbb F_2$.

Solved by gpt-5.6-sol high.

Solution

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

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{title2=$\mathbb F_k$}
{wiki}

The Hirzebruch surface is the ruled surface
$$
\mathbb F_k=\mathbb P_{\mathbb P^1}(\mathcal O\oplus\mathcal O(k)).
$$
It is the smooth toric surface with rays $(1,0),(0,1),(-1,k),(0,-1)$.


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