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Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry

Summary: Beitr¨age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 34 (1993), No. 1, 119­150.
Deformations of Affine Torus Varieties
Klaus Altmann \Lambda
Fachbereich Mathematik, Humboldt­Universit¨at zu Berlin
Institut f¨ur reine Mathematik, Ziegelstr. 13a, D­10117 Berlin
1. Introduction
1.1 Based on the paper [6] of Koll'ar/ Shepherd­Barron, many investigations of the base
space of the versal deformation of a two dimensional cyclic quotient singularity were done
in the last years: In [2], J. Arndt gave an algorithm for computing the equations of the
base space; in [7] and [3], J. Stevens and J.A. Christophersen described the components of
the reduced base space in a more qualitative way using continued fractions.
Cyclic quotient singularities are exactly those singularities that appear as two dimen­
sional affine torus varieties:
X(n; q) (quotient of the ZZ= nZZ ­action ¸ 7!
\Gamma ¸ 0
0 ¸ q
\Delta on I
C 2 ) can be built as an affine


Source: Altmann, Klaus - Fachbereich Mathematik und Informatik & Institut für Mathematik, Freie Universität Berlin


Collections: Mathematics