Abstract
The hierarchy of formsf(dx 1∧...∧dx n )α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.
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Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 12, pp. 37–46, 1987.
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Arnol'd, V.I. Poisson structures on the plane and other powers of volume forms. J Math Sci 47, 2509–2516 (1989). https://doi.org/10.1007/BF01102994
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DOI: https://doi.org/10.1007/BF01102994