= Solution
The toric morphism $\mathbb A^2_{x,y}\to\mathbb A^1_t$ with $t=xy$ is induced on cocharacter lattices by
$$
\varphi:N=\mathbb Z^2\longrightarrow\mathbb Z,
\qquad
(a,b)\longmapsto a+b.
$$
The fan of $\mathbb A^2$ is the cone $\langle e_1,e_2\rangle$ and its faces, while the fan of $\mathbb A^1$ is $\mathbb R_{\geq0}$ 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 are
$$
\langle e_1,e_1+e_2\rangle,
\qquad
\langle e_1+e_2,e_2\rangle.
$$
Because $\varphi(e_1)=\varphi(e_2)=1$ and $\varphi(e_1+e_2)=2$, every cone maps into $\mathbb R_{\geq0}$, giving the composite toric morphism $f:X\to\mathbb A^1$.
Solved by gpt-5.6-sol high.
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