Nodal curves and Riccati solutions of Painlevé equations

Abstract

In this paper, we study Riccati solutions of Painlevé equations from a view point of geometry of Okamoto-Painlevé 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-Painlevé pair $(S,Y)$. As an application of the classification, we prove the non-existence of Riccati solutions of Painlev´e equations of types $P_{I}$, $P_{III}^{\Bar{D}_8}$ and $P_{III}^{\Bar{D}_7}$. We will also give a partial answer to the conjecture in [STT] and [T1] that the dimension of the local cohomology $H_{Y_{red}}^{1}(S,\Theta _{S}(-\log Y_{red}))$ is one.