Let
$$
v=e_1+e_2+e_3
$$
and let $\widetilde\Sigma$ be the <star subdivision> of $\sigma$ through the ray $\mathbb R_{\geq0}v$. Its maximal cones are
$$
\operatorname{Cone}(v,e_2,e_3),\qquad
\operatorname{Cone}(e_1,v,e_3),\qquad
\operatorname{Cone}(e_1,e_2,v).
$$
Since $\widehat\varphi(v)=0$, their images lie respectively in
$$
\operatorname{Cone}(f_2,-f_1-f_2),\qquad
\operatorname{Cone}(f_1,-f_1-f_2),\qquad
\operatorname{Cone}(f_1,f_2).
$$
Thus $\widehat\varphi$ is a morphism of fans from $\widetilde\Sigma$ to $\Sigma$.

The <toric blowup at a torus-fixed point> identifies
$$
X_{\widetilde\Sigma}\cong\operatorname{Bl}_0\mathbb A^3.
$$
More explicitly, the <blowup of affine space at the origin> is the total space of $\mathcal O_{\mathbb P^2}(-1)$. The induced <toric morphism>
$$
g:X_{\widetilde\Sigma}\longrightarrow X_\Sigma\cong\mathbb P^2
$$
is 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 $f$.

Solution

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

{title2=$\operatorname{Bl}_0\mathbb A^n$}

The blowup of $\mathbb A^n$ at the origin is the total space of the tautological line bundle $\mathcal O_{\mathbb P^{n-1}}(-1)$. Its exceptional divisor is the zero section $\mathbb P^{n-1}$, and projection to the direction of a line gives the bundle map to $\mathbb P^{n-1}$.


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 Incoming links: Blowup of affine space at the origin