Abstract
This paper contains some contributions to a refined study of linear actions of the group G=Sl(2,ℂ) on a vectorspace V. More precisely, it deals with the study of the sophisticated geometry of orbitclosures. The case where V=Rn is the irreducible G-module of binary n-forms is of particular importance, since any finite-dimensional G-module is a direct sum of such irreducible G-modules. Hadžiev 1966 [Ha1] started investigating orbitclosures of binary n-forms according to their dimension and their decomposition into G-orbits. His results were generalized to orbitclosures in arbitrary finite-dimensional G-modules by Popov 1973 [P1] and [P2], who also began the classification of G-orbitclosures up to G-isomorphism. An intrinsic definition of orbitclosures required for such a classification is the following: An algebraic variety with a regular action of the group G, briefly a G-variety, is called quasihomogeneous if it contains a dense orbit. Popov 1973 [P1] classified all normal affine quasihomogeneous G-varieties. In addition, Luna and Vust 1983 [LV] classified the normal (not necessarily affine) quasihomogeneous G-varieties. These classifications use discrete parameters, which even in the most complicated affine case are rather simple, namely a natural and a rational number. This paper deals with the investigation of non-normal affine quasi-homogeneous G-varieties. In particular, numerical invariants for G-isomorphism are obtained, which are of an essentially higher complexity than in the normal case. Moreover, it is shown that the classification involves continuous parameters and modulispaces of arbitrary high dimension can occur.
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Literaturverzeichnis
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Bartels, D. (1985). Quasihomogene affine Varietäten für SL(2,ℂ). In: Malliavin, MP. (eds) Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin. Lecture Notes in Mathematics, vol 1146. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074535
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DOI: https://doi.org/10.1007/BFb0074535
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