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

Take the weighted projective plane $X=\mathbb{P}(1,1,2)$. Its fan in $N={\mathbb{Z}}^2$ has rays generated byand all three two-dimensional cones spanned by adjacent rays. Their union is $N_{\mathbb{R}}$, so the fan is complete and the toric variety $X$ is proper.

The determinants of the cones $⟨v_1,v_2⟩$ and $⟨v_2,v_0⟩$ have absolute value one, whileThe smoothness criterion for a toric variety therefore shows that the affine chart for $⟨v_0,v_1⟩$ is singular; it is the cyclic quotient singularity of type $\frac{1}{2}(1,1)$. Thus $X$ is a singular proper toric surface.

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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 ${\mathbb{F}}_2$.

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On ${\mathbb{F}}_2$, let $S$ be the negative section of a Hirzebruch surface, so $S^2=-2$, and let $F$ be the fiber class of a Hirzebruch surface, with $F^2=0$ and $S\cdot F=1$. The line bundle $\mathcal{O}(S+3F)$ is very ample. Indeed, the toric ampleness criterion for $aS+bF$ isand on a smooth complete toric variety every ample line bundle is very ample. These inequalities hold for $(a,b)=(1,3)$.

Explicitly, after choosing the standard lattice coordinates for ${\mathbb{F}}_2$, its lattice polytope of a toric divisor isThe monomials indexed by the lattice points of this polygon separate torus orbits and tangent directions, so their Kodaira map is a closed embedding. This directly verifies that the resolved proper toric surface is projective.

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