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

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 $f:X\to {\mathbb{A}}^1$.

Solved by gpt-5.6-sol high.

For a torus character ${\chi }^m$, its principal divisor on a toric variety isHere $t=xy={\chi }^{(1,1)}$, so on the blown-up fanThe scheme-theoretic fiber $X_0=V(f^{\ast }t)$ therefore contains the exceptional divisor $D_{e_1+e_2}$ with multiplicity two. Its defining ideal is not radical along that divisor, so $X_0$ is a nonreduced scheme.

Solved by gpt-5.6-sol high.

The two standard affine charts of the blowup restrict to an affine open cover of $X_0$. On the $x$-chart write $y=xu$. Since $xy=x^{2}u$, the fiber isOn the $y$-chart write $x=yv$, givingOn their overlap, $u$ and $v$ are invertible with $v=u^{-1}$ and $y=xu$. The squared factors $x^2$ and $y^2$ display the doubled exceptional component explicitly.

Solved by gpt-5.6-sol high.