Abstract. In this paper, we introduce the notion of generalized rational Okamoto-Painlevé pair (S, Y ) by generalizing the notion of the spaces of initial conditions of Painlevé equations. After classifying those pairs, we will establish an algebro-geometric approach to derive the Painlevé differential equations from the deformation of Okamoto-Painlevé pairs by using the local cohomology groups. Moreover the reason why the Painlevé equations can be written in Hamiltonian systems is clarified by means of the holomorphic symplectic structure on S − Y . Hamiltonian structures for Okamoto-Painlevé pairs of typeẼ 7 (= P II ) andD 8 (= PD 8 III ) are calculated explicitly as examples of our theory.

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 PI , PD 8 III and PD 7 III . 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, ΘS(− log Y red )) is one.1991 Mathematics Subject Classification. 14D15, 34M55, 32G10.

In [as reported by Saito et al. (J. Algebraic Geom. 11:311-362, 2002)], generalized Okamoto-Painlevé pairs are introduced as a generalization of Okamoto's space of initial conditions of Painlevé equations (cf. [Okamoto (Jpn. J. Math. 5:1-79, 1979)]) and we established a way to derive differential equations from generalized rational Okamoto-Painlevé pairs through deformation theory of nonsingular pairs. In this article, we apply the method to concrete families of generalized rational Okamoto-Painlevé pairs with given affine coordinate systems and for all eight types of such Okamoto-Painlvé pairs we write down Painlevé equations in the coordinate systems explicitly. Moreover, except for a few cases, Hamitonians associated to these Painlevé equations are also given in all coordinate charts.

In the theory of deformation of Okamoto-Painlevé pair (S, Y ), a local cohomology group H 1 D (Θ S (− log D)) plays an important role. In this paper, we estimate the local cohomology group of pair (S, Y ) for several types, and obtain the following results. For a pair (S, Y ) corresponding to the space of initial conditions of the Painlevé equations, we show that the local cohomology group H 1 D (Θ S (− log D)) is at least 1 dimensional. This fact is the key to understand Painlevé equation related to (S, Y ). Moreover we show that, for the pairs (S, Y ) of typeà 8 , the local cohomology group H 1 D (Θ S (− log D)) vanish. Therefore in this case, there is no differential equation on S − Y in the sense of the theory.1991 Mathematics Subject Classification. 14D15, 14J26, 32G10, 34M55.