Nodal curves and Riccati solutions of Painlevé equations
The authors study the Riccati solutions of the Painlevé equation as geometrical interpretations of Okamoto-Painlevé pair \((S, Y)\), a projective smooth surface \(S\) and effective divisors \(Y\) on \(S\). After establishing the correspondence between (rational) nodal curves on \(S-Y\) and Riccati solutions, the authors give the complete classification of the configurations of the curves for each \((S, Y)\). As its application, the authors prove the non-existence of Riccati solutions of Painlevé equations of type \(P_I\), \(P_{II}^{\widetilde{D}_8}\), \(P_{III}^{\widetilde{D}_7}\) and provide a partial answer to a conjecture of \textit{M. Saito T. Takebe} and \textit{H. Terajima} [J. Algebr. Geom. 11, No. 2, 311--362 (2002; Zbl 1022.34079)] and \textit{H. Terajima} [Jap. J. Math., New Ser. 28, No. 1, 137--162 (2002; Zbl 1054.14005)] on the dimension of a local cohomology associated with \((S, Y)\).
scientific article; zbMATH DE number 5010263