Nodal curves and Riccati solutions of Painlev\'e equations

In this paper, we study Riccati solutions of Painlev'e equations from a view point of geometry of Okamoto-Painlev'e pairs $(S,Y)$. After establishing the correspondence between (rational) nodal curves on $S-Y$ and Riccati solutions, we give the complete classification of the configurations of nodal curves on $S-Y$ for each Okamoto-Painlev'e pair $(S,Y)$. As an application of the classification, we prove the non-existence of Riccati solutions of Painlev'e equations of types $P_I,P_{III}^{{\tilde{D}}_8}$ and $P_{III}^{{\tilde{D}}_7}$. We will also give a partial answer to the conjecture in (STT) and (T) that the dimension of the local cohomology $H_{Y_{red}}^1(S,{\mathrm{\Theta }}_S(-\log Y_{red}))$ is one.