Nodal curves and Riccati solutions of Painlevé equations
Loading...
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
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 $P_{I}, P_{III}^{\tilde{D}_8}$ 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 $H^1_{Y_{red}}(S,Θ_S(-\log Y_{red}))$ is one.
30 pages, 4 figures
30 pages, 4 figures