Hyperelliptic Integrable Systems on K3 and Rational Surfaces

Details Hyperelliptic Integrable Systems on K3 and Rational Surfaces Hyperelliptic Integrable Systems on K3 and Rational Surfaces

2000-07-12

arxiv.org/abs/math/0007073v2

Phys. Lett. A283 (2001), 201--208

Mathematics

Algebraic Geometry

LaTeX2e using packages "amsmath,amssymb", 15 pages, no figure

Scientific paper

10.1016/S0375-9601(01)00241-9

We show several examples of integrable systems related to special K3 and rational surfaces (e.g., an elliptic K3 surface, a K3 surface given by a double covering of the projective plane, a rational elliptic surface, etc.). The construction, based on Beauvilles's general idea, is considerably simplified by the fact that all examples are described by hyperelliptic curves and Jacobians. This also enables to compare these integrable systems with more classical integrable systems, such as the Neumann system and the periodic Toda chain, which are also associated with rational surfaces. A delicate difference between the cases of K3 and of rational surfaces is pointed out therein.

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