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| In [[algebraic geometry]], the '''Koras–Russell cubic threefolds''' are smooth affine contractible threefolds studied by {{harvtxt|Koras|Russell|1997}} that have a hyperbolic action of a one-dimensional [[torus]] with a unique fixed point, such that the quotients of the threefold and the tangent space of the fixed point by this action are isomorphic.
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| Much earlier than the above referred paper, Russell noticed that the hypersurface <math>x+x^2y+z^2+t^3=0</math> has properties very similar to the affine 3-space like contractibility and was interested in distinguishing it from the affine 3-space as algebraic varieties, necessary for linearizing <math>\mathbf{C}^*</math> actions on <math>\mathbf{A}^3</math>. This led Makar-Limanov to the discovery of an invariant, later called the [[ML-invariant]] of an affine variety. The ML-invariant was successfully used to distinguish the Russell cubic from the affine 3-space among its many other successes. In the paper above, Koras and Russell look at a large family of smooth contractible hypersurfaces which contains the Russell cubic as a special case.
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| ==References==
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| *{{Citation | last1=Koras | first1=M. | last2=Russell | first2=Peter | title=Contractible threefolds and C<sup>*</sup>-actions on C<sup>3</sup> | mr = 1487230 | zbl = 0882.14013 | year=1997 | journal=Journal of Algebraic Geometry | issn=1056-3911 | volume=6 | issue=4 | pages=671–695}}
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| {{DEFAULTSORT:Koras-Russell cubic threefold}}
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| [[Category:Threefolds]]
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Revision as of 00:17, 1 March 2014
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