On the Classification of Smooth Toric Surfaces with Exactly One Exceptional Curve

We classify all smooth projective toric surfaces containing exactly one exceptional curve. We show that every such surface 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 a sequence of blow-ups of length equal to the level of the rational number $\{r\}\in (0,1)$ in the classical Farey tree. 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 the two-dimensional regular fans $\Sigma _r$ of toric surfaces $S_r$ to construct two-dimensional colored fans $\Sigma _r^{\textup {c}}$ of minimal horospherical threefolds $V_r$ having a regular $(\mathrm {SL}(2)\times \mathbb G_{\textup {m}})$-action. The varieties $V_r$ are toric and minimal. Their classification was obtained by D. Guan. We establish a direct combinatorial connection between the three-dimensional fans $\widetilde {\Sigma }^{\textup {c}}_r$ of threefolds $V_r$ and the two-dimensional fans $\Sigma _r$ of surfaces $S_r$.