Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 152 / 2 / b

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 $f$.