Nodal curves and Riccati solutions of Painlevé equations

2002/01/23 by Masa-Hiko Saito, Hitomi Terajima
Mathematics · #math.AG #math.QA #msc:14D15 #msc:34M55 #msc:32G10

paper · pdf

published as J. Math. Kyoto Univ., 44(2004), no.3, 529--568 · 30 pages, 4 figures

arxiv created 2002/01/23 · arxiv updated 2009/11/30

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 Painlevé equations of types P_I, P_III^D₈ and P_III^D₇. We will also give a partial answer to the conjecture in (STT) and (T) that the dimension of the local cohomology H¹_Y_red(S,Θ_S(-log Y_red)) is one.

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