On the classification of 3-dimensional \(SL_2 (\mathbb{C})\)-varieties. (English) Zbl 0964.14042

Kebekus, Stefan

Nagoya Math. J. 157, 129-147 (2000).

In the first part of the paper under review 3-dimensional complex \(Sl_2\)-varieties are considered where the generic \(Sl_2\)-orbit is a surface. The main result is an elementary criterion for fibers of the categorical quotient map to be irreducible or normal. A fine classification of neighborhoods of reduced fibers is given. These results are employed in the main part of the paper to yield a classification of certain 3-dimensional varieties with \(Sl_2\)-action. These varieties appear if one applies the minimal model program to arbitrary projective 3-folds with \(Sl_2\)-action. Their classification is an ingredient of a more general classification program which was considered by S. Kebekus in the preceding paper [Nagoya Math. J. 157, 149-176 (2000; see the preceding review Zbl 0694.14041)].

Reviewer: Stefan Kebekus (Bayreuth)

MSC:

14M17	Homogeneous spaces and generalizations
14J30	\(3\)-folds
14L30	Group actions on varieties or schemes (quotients)
32M12	Almost homogeneous manifolds and spaces
14E30	Minimal model program (Mori theory, extremal rays)

Keywords:

3-dimensional complex \(Sl_2\)-varieties; quotient map to; reduced fibers; minimal model program

Citations:

Zbl 0694.14041

References:

[1]	(1998)
[2]	Aspekte der Mathematik, D1, Vieweg 2 (1985)
[3]	Classification Theorems for Almost Homogeneous Spaces (1980)
[4]	Advanced studies in pure mathematics 2 (1962)
[5]	Introduction to Toric Varieties 131 (1993)
[6]	Proceedings of Symposia in Pure Mathematics 46 pp 345– (1987)
[7]	Commutative Algebra with a View Toward Algebraic Geometry (1995) · Zbl 0819.13001
[8]	Algebraic Geometry Angers (1979)
[9]	Hyperplane Sections and Deformations (1982)
[10]	DOI: 10.1016/0021-8693(74)90194-X · Zbl 0284.14009 · doi:10.1016/0021-8693(74)90194-X
[11]	Nagoya Math. J 86 pp 155– (1982) · Zbl 0445.14002 · doi:10.1017/S0027763000019838
[12]	In M. Raynaud and T. Shioda, editors, Algebraic geometry, Proc. Jap.-Fr. Conf., Tokyo and Kyoto 1982, Lect. Notes Math 1016 pp 490– (1983)
[13]	Izwestiya RAN 61 pp 685– (1998)
[14]	Nagoya Math. J 98 pp 43– (1985) · Zbl 0589.14005 · doi:10.1017/S0027763000021358
[15]	DOI: 10.2307/2007050 · Zbl 0557.14021 · doi:10.2307/2007050
[16]	DOI: 10.1007/BF02564633 · Zbl 0545.14010 · doi:10.1007/BF02564633
[17]	Expo. Math 4 (1986)