Abstract
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 \(\Sigma_r^{\text{c}}\) of minimal horospherical threefolds \(V_r\) having a regular \((\mathrm{SL}(2)\times\mathbb G_{\text{m}})\) -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 \(\widetilde{\Sigma}{}^{\text{c}}_r\) of threefolds \(V_r\) and the two-dimensional fans \(\Sigma_r\) of surfaces \(S_r\) .
