On the classification of smooth toric surfaces with exactly one exceptional curve

We classify all smooth projective toric surfaces $S$ containing exactly one exceptional curve. We show that every such surface $S$ is isomorphic to either $\mathbb{F}_1$ or a surface $S_r$ defined by a rational number $r \in \mathbb{Q} \setminus \mathbb{Z}$ $(r >1)$. If $a:= [ r]$ then $S_r$ is obtained from the minimal desingularization of the weighted projective plane $\mathbb{P}(1, 2, 2a+1)$ by toric blow-ups whose quantity equals the level of the rational number $\{ r \} \in (0,1)$ in the classical Farey tree. Moreover, we show that if $r = b/c$ with coprime $b$ and $c$, then $S_r$ is the minimal desingularization of the weighted projective plane $\mathbb{P}(1, c, b)$. We apply $2$-dimensional regular fans $\Sigma_r$ of toric surfaces $S_r$ for constructing $2$-dimensional colored fans $\Sigma^c$ of minimal horospherical $3$-folds having a regular $SL(2) \times \mathbb{G}_m$-action. The latter are minimal toric $3$-folds $V_r$ classified by Z. Guan. We establish a direct combinatorial connection between the $3$-dimensional fans $\widetilde{\Sigma}^c_r$ of $3$-folds $V_r$ and the $2$-dimensional fans $\Sigma_r$ of surfaces $S_r$.