Nodal curves and Riccati solutions of Painlevé equations
type: Item
instance of: scholarly article
title: Nodal curves and Riccati solutions of Painlevé equations
zbMATH Open document ID: 1117.14015
author: Masa-Hiko Saito
author: Hitomi Terajima
published in: Journal of Mathematics of Kyoto University
publication date: 2006-03-09T00:00:00Z
review text: 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 \((S, Y)\).
reviewed by: Shigeki Matsutani
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 14H70
Mathematics Subject Classification ID: 32G10
zbMATH DE Number: 5010263
zbMATH Keywords: Painlevé equation
zbMATH Keywords: Riccati solution
zbMATH Keywords: Okamoto-Painlevé pair
zbMATH Keywords: nodal curve
zbMATH Keywords: local cohomology
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250283083
OpenAlex ID: W1530166346
DOI: 10.1215/KJM/1250283083
description: scientific article; zbMATH DE number 5010263