Nodal curves and Riccati solutions of Painlevé equations

الأصل · EN

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 Painlevé equations of types Pᵢ, PIIID₈ and PIIID₇. We will also give a partial answer to the conjecture in (STT) and (T) that the dimension of the local cohomology H¹Yred(S,Θₛ(- Yred)) is one.