Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 152 (Toric Varieties)

Let $v_1,\dots ,v_r$ be the primitive lattice generators of the rays of the strongly convex rational cone $\sigma \subseteq N_{\mathbb{R}}$. The smoothness criterion for a toric variety says that the affine toric variety $U_{\sigma }$ is smooth exactly when $v_1,\dots ,v_r$ form part of a $\mathbb{Z}$-basis of the cocharacter lattice of an algebraic torus $N$. Equivalently, there is a basis $e_1,\dots ,e_n$ of $N$ such thatIn particular, the cone must be a simplicial polyhedral cone and its ray generators must be primitive; for a full-dimensional cone the criterion says that those generators form a lattice basis.

The orbit-cone correspondence is an inclusion-reversing bijectionIf $n=\mathrm{rank}N$, thenThe affine chart and orbit closure in a toric variety areIn particular, the zero cone corresponds to the dense algebraic torus, while maximal cones correspond to torus-fixed points.

For every $t\in T_N$, the torus action mapis an automorphism of $X_{\Sigma }$. It induces isomorphisms of local rings, so it carries smooth points to smooth points and singular points to singular points. The singular locus is consequently invariant under $T_N$. If it contains one point $x$, it contains the entire orbit $T_N\cdot x$; hence the torus-invariance of the singular locus of a toric variety proves that $X_{\Sigma }^{\mathrm{sing}}$ is a union of the orbits in the orbit-cone correspondence.

Write $m=a{e}_1^{\ast }+b{e}_2^{\ast }\in M$. The inequalities defining the dual cone areThusIts Hilbert basis of a rational cone isIndeed, after subtracting copies of $(2,3)$ one reduces to $b=0,1,$ or $2$, and the remaining point is generated by $(1,0)$ and $(1,1)$. The coordinate ring of an affine toric variety is thereforePutting $A=x$, $B=xy$, and $C=x^2{y}^3$ gives the alternative presentation

The primitive ray generators of $\sigma$ areand $|\det (v_0,v_3)|=3$. They do not form a lattice basis, so the smoothness criterion for a toric variety shows that $U_{\sigma }$ is singular. The presentation from part (i) gives the same conclusion by the Jacobian criterion: all derivatives of $B^3-AC$ vanish at $A=B=C=0$.

Insert the primitive raysLet $\Sigma$ consist of the conesand all their faces. Sinceevery cone of $\Sigma$ is smooth. Its support is $\sigma$, so this fan subdivision induces a proper birational toric morphismThe source is smooth, and $p$ is an isomorphism over the dense algebraic torus; it is therefore a toric resolution of singularities.

Let $D_1$ and $D_2$ be the toric divisors corresponding respectively to the primitive ray generators $v_1=(3,-2)$ and $v_2=(0,1)$ of the original cone. The principal divisor on a toric variety formula applied to the characters $x={\chi }^{e_1^{\ast }}$ and $y={\chi }^{e_2^{\ast }}$ givesThe toric divisor class sequence therefore presents the divisor class group asThe class of $D_1$ is a generator and $[D_2]=2[D_1]$.

The product fan for ${\mathbb{P}}^1\times {\mathbb{A}}^1$ has rays generated byand maximal cones $\mathrm{Cone}(e_1,e_2)$ and $\mathrm{Cone}(e_2,-e_1)$. The given fan is obtained by the star subdivision of the first maximal cone throughBy the toric blowup at a torus-fixed point, this subdivision blows up the torus-fixed point corresponding to $\mathrm{Cone}(e_1,e_2)$. Hencefor that torus-fixed point $p$.

The fan of the affine line consists of the zero cone and ${\mathbb{R}}_{\ge 0}{e}^{\prime }$. Under $\phi (e_1)=0$ and $\phi (e_2)=e^{\prime }$, the four primitive source rays map asEvery cone of the source fan consequently maps into ${\mathbb{R}}_{\ge 0}{e}^{\prime }$. The defining criterion for a toric morphism is satisfied, so $\phi$ inducesUnder the blowup description from part (i), this is the blowup map followed by the projection ${\mathbb{P}}^1\times {\mathbb{A}}^1\to {\mathbb{A}}^1$.

Before blowing up, the fiber over zero is the smooth curveand the blowup center $p$ lies on it. Its total transform is the union of the strict transform ${\tilde{F}}_0$ and the exceptional divisor $E$. Blowing up a smooth point of a smooth curve does not change the curve, so ${\tilde{F}}_0\cong {\mathbb{P}}^1$, while $E\cong {\mathbb{P}}^1$. They meet transversely in the single point recording the tangent direction of $F_0$ at $p$. Thereforeis two copies of the complex projective line meeting at one point.

The inverse image of the dense torus ${\mathbb{A}}^1\setminus \{0\}={\mathbb{G}}_m$ is represented by the subfan formed from cones mapping to the zero cone of the target fan. These areThis is the fan of the projective line in the $e_1$-axis, together with the zero fan in the complementary $e_2$-direction. By the product fan construction,It also follows geometrically because the blowup center lies over zero, so the blowup is unchanged over ${\mathbb{A}}^1\setminus \{0\}$.

Part (iii) givestwo complex projective lines meeting at one point. If $x\ne 0$, the fiber ${\mathbb{P}}^1\times \{x\}$ misses the blowup center. The blowup map is an isomorphism of algebraic varieties away from its center, sofor every nonzero closed point $x\in {\mathbb{A}}^1$.

The cone $\sigma =\mathrm{Cone}(e_1,e_2,e_3)$ and all its faces form the standard fan of affine three-space, soThe complete target fan has primitive raysand every cone generated by a proper subset of them. This is the two-dimensional simplex fan, hence

The images of the three rays of $\sigma$ areNo cone of the complete fan of ${\mathbb{P}}^2$ contains all three rays, so $\hat{\phi }(\sigma )$ is not contained in one target cone. Thus $\hat{\phi }$ is not a morphism of fans and does not define a regular toric morphism on all of $U_{\sigma }$.

On the dense algebraic torus, the pullbacks of the two standard target characters areUnder the standard torus chart $[u:v:1]\in {\mathbb{P}}^2$, the resulting rational map of toric varieties from a lattice homomorphism isIts homogeneous coordinates vanish simultaneously exactly at the origin. Hence its indeterminacy locus is $\{0\}$.

Letand let $\tilde{\Sigma }$ be the star subdivision of $\sigma$ through the ray ${\mathbb{R}}_{\ge 0}v$. Its maximal cones areSince $\hat{\phi }(v)=0$, their images lie respectively inThus $\hat{\phi }$ is a morphism of fans from $\tilde{\Sigma }$ to $\Sigma$.

The toric blowup at a torus-fixed point identifiesMore explicitly, the blowup of affine space at the origin is the total space of ${\mathcal{O}}_{{\mathbb{P}}^2}(-1)$. The induced toric morphismis its bundle projection: it sends a nonzero vector to its direction $[x_1:x_2:x_3]$ and restricts to the identity on the exceptional divisor ${\mathbb{P}}^2$. Therefore $g$ resolves the indeterminacy locus of $f$.