Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 (Toric Geometry)

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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Write the invariant prime divisors in the order of the raysas $D_1,\dots ,D_4$. The toric divisor class sequence isand the two characters in the standard basis of $M$ giveThuswhere $S=D_2$, $F=D_1=D_3$, and $D_4=S+kF$. Since the fan is smooth, this is also the Picard group.

The orbit-cone correspondence decomposes ${\mathbb{F}}_k$ into one two-dimensional torus, four one-dimensional torus orbits, and four fixed points. Only the zero-dimensional orbits contribute to the compactly supported Euler characteristic, so

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The Cox construction uses coordinates $x_1,\dots ,x_4$ of degreesIts irrelevant locus isso put $U={\mathbb{A}}^4\setminus (V(x_1,x_3)\cup V(x_2,x_4))$. The algebraic torus $G=({\mathbb{G}}_m{)}^2$ acts byEvery point of $U$ has a nonzero coordinate from each opposing pair, so the action has the expected closed orbits and trivial stabilizers. The geometric quotient is

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Take the invariant divisor $D_1$, whose class is $F$. Its global sections are represented in the Cox ring by $x_1$ and $x_3$. They have no common zero on $U$, so $F$ is basepoint-free. Its Kodaira map isThis is exactly the ruling ${\mathbb{F}}_k={\mathbb{P}}_{{\mathbb{P}}^1}(\mathcal{O}\oplus \mathcal{O}(k))\to {\mathbb{P}}^1$, and its fibers have divisor class $F$.

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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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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.

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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.

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Let $q:N_{\mathbb{R}}\to N(\tau {)}_{\mathbb{R}}=N_{\mathbb{R}}/\mathrm{span}(\tau )$ be the quotient map. The images $q(\sigma )$ for $\tau ⪯\sigma$ satisfy the face and intersection axioms, so $\mathrm{Star}(\tau )$ is a fan.

To prove completeness, take $\bar{v}\in N(\tau {)}_{\mathbb{R}}$, choose a lift $v$, and choose $w$ in the relative interior of $\tau$. Since $\Sigma$ is complete, each $v+mw$ lies in some cone of $\Sigma$. There are finitely many cones, so one cone $\sigma$ contains $v+m_{j}w$ for an unbounded sequence $m_j$. Closedness gives $w\in \sigma$. Because $w$ lies in the relative interior of the cone $\tau$ and $\sigma \cap \tau$ is a face of $\tau$, this forces $\tau ⪯\sigma$. Finally,Thus every point of the quotient lies in the support of $\mathrm{Star}(\tau )$, proving that it is a complete fan.

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Let a complete fan in a $d$-dimensional real vector space have ray generators $v_1,\dots ,v_r$. Every cone of the fan lies in the positive hull $\mathrm{cone}(v_1,\dots ,v_r)$. If $r<d$, these vectors do not span the space. If $r=d$, either they still fail to span, or they are linearly independent and their positive hull is a proper strictly convex cone. In either case their positive hull cannot be the whole vector space, contrary to completeness. Hence $r\ge d+1$.

The bound is attained: take rays throughand cones generated by every proper subset of these rays. This is the simplex fan, which is complete and has $d+1$ rays. Therefore the minimal number is $d+1=\dim N_{\mathbb{R}}+1$.

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