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Nodal curves and Riccati solutions of Painlevé equations | | Masa-Hiko Saito
; Hitomi Terajima
; | | Date: |
23 Dec 2001 | | Journal: | J. Math. Kyoto Univ., 44(2004), no.3, 529--568 | | Subject: | Algebraic Geometry; Quantum Algebra MSC-class: 14D15, 34M55, 32G10 | math.AG math.QA | | Affiliation: | Kobe University | | 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 $P_{I}, P_{III}^{ ilde{D}_8}$ and $P_{III}^{ ilde{D}_7}$. We will also give a partial answer to the conjecture in (STT) and (T) that the dimension of the local cohomology $H^1_{Y_{red}}(S,Theta_S(-log Y_{red}))$ is one. | | Source: | arXiv, math.AG/0201225 | | Services: | Forum | Review | PDF | Favorites | |
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