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Classification(s) of Danielewski hypersurfaces

Poloni, P.-M

In: Transformation Groups, 2011, vol. 16, no. 2, p. 579-597

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    Title
    • Classification(s) of Danielewski hypersurfaces
    Author
    • Poloni, P.-M. Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051, Basel, Switzerland
    Document Type
    • Postprint
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:310397
    Summary
    The Danielewski hypersurfaces are the hypersurfaces X Q,n in $ {\mathbb{C}^3} $ defined by an equation of the form x n y = Q(x, z) where n ⩾ 1 and Q(x, z) is a polynomial such that Q(0, z) is of degree at least two. They were studied by many authors during the last twenty years. In the present article, we give their classification as algebraic varieties. We also give their classification up to automorphism of the ambient space. As a corollary, we obtain that every Danielewski hypersurface X Q,n with n ⩾ 2 admits at least two nonequivalent embeddings into $ {\mathbb{C}^3} $