Transactions of the American Mathematical Society

Projective threefolds on which $\mathbf{SL}(2)$ acts with 2-dimensional general orbits

Author(s): T. Nakano
Journal: Trans. Amer. Math. Soc. 350 (1998), 2903-2924.
MSC (1991): Primary 14L30; Secondary 14E30
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Abstract: The birational geometry of projective threefolds on which $\mathbf{SL}(2)$ acts with 2-dimensional general orbits is studied from the viewpoint of the minimal model theory of projective threefolds. These threefolds are closely related to the minimal rational threefolds classified by Enriques, Fano and Umemura. The main results are (i) the $\mathbf{SL}(2)$ -birational classification of such threefolds and (ii) the classification of relatively minimal models in the fixed point free cases.

[A]: M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Etudes Sci. Publ. Math. 36 (1969), 23-58. MR 42:3087
[B]: A. Bialynicki-Birula, On action of $\mathbf{SL}(2)$ on complete algebraic varieties, Pacific J. Math. 86 (1980), 53-58. MR 81i:14031
[CKM]: H. Clemens, J. Kollár, and S. Mori, Higher dimensional complex geometry, Astérisque 166, Société Mathématique de France, 1988. MR 90j:14046
[De]: M. Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. École Norm. Sup. (4e) 3 (1970), 507-588. MR 44:1672
[Du1]: E. Duma, Algebraic $\mathbf{SL}(2)$ -actions: Deformations of infinite isotropy subgroups, Colloq. Math. 58 (1990), 221-231. MR 91g:14042
[Du2]: -, On $\mathbf{SL}(2)$ -actions without $3$ -dimensional orbits, Colloq. Math. 58 (1990), 231-241. MR 91f:14043
[EF]: F. Enriques and G. Fano, Sui gruppi di transformazioni cremoniane dello spazio, Ann. Mat. Pura Appl. 15 (1897), 59-98.
[F]: G. Fano, I gruppi di Jonquiéres generalizzati, Atti della R. Accad. di Troino 33 (1898), 221-271.
[FG]: W. Fischer and H. Grauert, Local-triviale Familien kompakter komplexer Mannigfaltigkeiten, Nachr. Akad. Wiss. Göttingen Math.-Phys. K1 II (1965), 89-94. MR 32:1731
[H]: R. Hermann, Lie groups: History, frontiers and applications, Vol. 1 (Sophus Lie's 1880 Transformation Group Paper), Mat. Sci. Press, Brookline, 1975.
[K]: H. Kraft, Algebraic automorphisms of affine space, Topological Methods in Algebraic Transformations Groups, H. Kraft et al.(eds.), Progress in Math. 80, Birkhäuser, Boston, Basel, Berlin, 1989, pp. 81-105. MR 91g:14044
[L]: D. Luna, Slices étales, Bull. Soc. Math. France Mem. 33 (1973), 81-105. MR 49:7269
[LV]: D. Luna and Th. Vust, Plongements d'espaces homogènes, Comment Math. Helv. 58 (1983), 186-245. MR 85a:14035
[Mab]: T. Mabuchi, On the classification of essentially effective $\mathbf{SL}(n,\mathbf C)$ -actions on algebraic $n$ -folds, Osaka J. Math. 16 (1979), 745-759. MR 81k:14033b
[Mat]: H. Matsumura, On algebraic groups of birational transformations, Rend. della Acad. Naz. del Lincei 34 (1963), 151-155. MR 28:3041
[Mo]: S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982), 133-176. MR 84e:14032
[Mu]: S. Mukai and H. Umemura, Minimal rational threefolds, Lecture Notes in Math., vol. 1016, M. Raynaud et al. (eds.), Springer-Verlag, Berlin, Heidelberg, New York, 1983, pp. 490-518. MR 85c:14027
[N1]: T. Nakano, Regular actions of simple algebraic groups on projective threefolds, Nagoya Math. J. 116 (1989), 139-148. MR 91b:14054
[N2]: -, On equivariant completions of $3$ -dimensional homogeneous spaces of $\mathbf{SL}(2,\mathbf C)$ , Japan. J. Math. 15 (1989), 221-273. MR 91b:14053
[N3]: -, Regular actions of semi-simple algebraic groups on projective threefolds, especially $\mathbf{SL}(2)$ , Group actions and invariant theory, A. Bialynicki-Birula et al. (eds.), CMS Conf. Proc. 10, AMS, Providence, Rhode Island, 1989, pp. 137-155. MR 90i:14051
[P]: V. L. Popov, Quasi-homogeneous affine algebraic varieties of the group $\mathbf{SL}(2)$ , Math. USSR Izv. 7 (1973), 793-831. MR 49:5018
[R]: M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401-443. MR 18:514a
[S]: H. Sumihiro, Equivariant completion II, J. Math. Kyoto Univ. 15 (1975), 573-605. MR 52:8137
[U1]: H. Umemura, Sur les sous-groupes algébriques primitifs du groupe de Cremona à trois variables, Nagoya Math. J. 79 (1980), 47-67. MR 81k:14011
[U2]: -, Maximal algebraic subgroups of the Cremona group of three variables (Imprimitive algebraic subgroups of exceptional type), Nagoya Math. J. 87 (1982), 59-78. MR 84b:14005
[U3]: -, On the maximal connected algebraic subgroups of the Cremona group II, Algebraic Groups and Related Topics, Advanced Studies in Pure Math. 6, Kinokuniya and North-Holland, Tokyo, Amsterdam, New York, Oxford, 1985, pp. 349-436. MR 87d:14008
[U4]: -, Minimal rational threefolds II, Nagoya Math. J. 110 (1988), 15-80. MR 89f:14040

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 14L30, 14E30

T. Nakano
Affiliation: Department of Mathematical Sciences, College of Science and Engineering, Tokyo Denki University, Hatoyama-machi, Hiki-gun, Saitama-ken, 350-0394, Japan
Email: nakano@r.dendai.ac.jp

DOI: 10.1090/S0002-9947-98-02081-9
PII: S 0002-9947(98)02081-9
Keywords: $\mathbf{SL}(2)$-action, projective threefolds, minimal models
Received by editor(s): July 20, 1996
Copyright of article: Copyright 1998, American Mathematical Society

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