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

Abstract

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 ${\tilde{\Sigma }}_r^c$ of $3$-folds $V_r$ and the $2$-dimensional fans ${\Sigma }_r$ of surfaces $S_r$.

16 pages, 12 figures