Abstract:
In this paper we prove that if S is a Poisson surface, i.e., a smooth algebraic surface with a Poisson structure, the Hilbert scheme of points of S has a natural Poisson structure, induced by the one of S. This generalizes previous results obtained by A. Beauville [B1] and S. Mukai [M2] in the symplectic case, i.e., when S is an abelian or K3 surface. Finally we apply our results to give some examples of integrable Hamiltonian systems naturally defined on these Hilbert schemes. In the simple case S=ℙ2 we obtain by this construction a large class of integrable systems, which includes the ones studied by P. Vanhaecke in [V1] and, more generally, in [V2].
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Received: 9 March 1998 / Revised version: 19 June 1998
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Bottacin, F. Poisson structures on Hilbert schemes of points¶of a surface and integrable systems. manuscripta math. 97, 517–527 (1998). https://doi.org/10.1007/s002290050118
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DOI: https://doi.org/10.1007/s002290050118