= Solution
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 $S\mathbin{\cdot}F=1$. The <line bundle> $\mathcal O(S+3F)$ is very ample. Indeed, the toric ampleness criterion for $aS+bF$ is
$$
a>0,\qquad b-2a>0,
$$
and on a smooth complete toric variety every ample line bundle is <very ample line bundle>[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> is
$$
P_{S+3F}=\operatorname{conv}\{(0,0),(3,0),(1,1),(0,1)\}.
$$
The 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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