Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 / 3 / a / Solution

The toric morphism ${\mathbb{A}}_{x,y}^2\to {\mathbb{A}}_t^1$ with $t=xy$ is induced on cocharacter lattices byThe fan of ${\mathbb{A}}^2$ is the cone $⟨e_1,e_2⟩$ and its faces, while the fan of ${\mathbb{A}}^1$ is ${\mathbb{R}}_{\ge 0}$ and its zero face. The blowup of the affine plane at the origin is the star subdivision obtained by inserting the ray through $e_1+e_2$; its maximal cones areBecause $\phi (e_1)=\phi (e_2)=1$ and $\phi (e_1+e_2)=2$, every cone maps into ${\mathbb{R}}_{\ge 0}$, giving the composite toric morphism $f:X\to {\mathbb{A}}^1$.

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