Projective threefolds on which 𝐒𝐋(2) acts with 2-dimensional general orbits

Nakano, T.

doi:10.1090/S0002-9947-98-02081-9

Projective threefolds on which $\mathbf {SL}(2)$ acts with 2-dimensional general orbits
HTML articles powered by AMS MathViewer

by T. Nakano PDF

Trans. Amer. Math. Soc. 350 (1998), 2903-2924 Request permission

Abstract:

The birational geometry of projective threefolds on which $\mathrm {SL}(2)$ 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 $\mathrm {SL}(2)$-birational classification of such threefolds and (ii) the classification of relatively minimal models in the fixed point free cases.

References

M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58. MR 268188
A. Białynicki-Birula, On action of $\textrm {SL}(2)$ on complete algebraic varieties, Pacific J. Math. 86 (1980), no. 1, 53–58. MR 586868
Herbert Clemens, János Kollár, and Shigefumi Mori, Higher-dimensional complex geometry, Astérisque 166 (1988), 144 pp. (1989) (English, with French summary). MR 1004926
Michel Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. École Norm. Sup. (4) 3 (1970), 507–588 (French). MR 284446
Ewa Duma, Algebraic $\textrm {SL}(2)$-actions: deformations of infinite isotropy subgroups, Colloq. Math. 58 (1990), no. 2, 221–231. MR 1060173, DOI 10.4064/cm-58-2-221-231
Ewa Duma, On $\textrm {SL}(2)$-actions without $3$-dimensional orbits, Colloq. Math. 58 (1990), no. 2, 233–241. MR 1060174, DOI 10.4064/cm-58-2-233-241
F. Enriques and G. Fano, Sui gruppi di transformazioni cremoniane dello spazio, Ann. Mat. Pura Appl. $(2^a)$ 15 (1897), 59–98.
G. Fano, I gruppi di Jonquiéres generalizzati, Atti della R. Accad. di Troino 33 (1898), 221–271.
Wolfgang Fischer and Hans Grauert, Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1965 (1965), 89–94 (German). MR 184258
R. Hermann, Lie groups: History, frontiers and applications, Vol. 1 (Sophus Lie’s 1880 Transformation Group Paper), Mat. Sci. Press, Brookline, 1975.
Hanspeter Kraft, Algebraic automorphisms of affine space, Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988) Progr. Math., vol. 80, Birkhäuser Boston, Boston, MA, 1989, pp. 81–105. MR 1040858
Domingo Luna, Slices étales, Sur les groupes algébriques, Bull. Soc. Math. France, Paris, Mémoire 33, Soc. Math. France, Paris, 1973, pp. 81–105 (French). MR 0342523, DOI 10.24033/msmf.110
D. Luna and Th. Vust, Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), no. 2, 186–245 (French). MR 705534, DOI 10.1007/BF02564633
Toshiki Mabuchi, On the classification of essentially effective $\textrm {SL}(2;\textbf {C})$ $\times \textrm {SL}(2;\textbf {C})$-actions on algebraic threefolds, Osaka Math. J. 16 (1979), no. 3, 727–744. MR 551585
Hideyuki Matsumura, On algebraic groups of birational transformations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 34 (1963), 151–155. MR 159825
Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176. MR 662120, DOI 10.2307/2007050
Shigeru Mukai and Hiroshi Umemura, Minimal rational threefolds, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 490–518. MR 726439, DOI 10.1007/BFb0099976
Tetsuo Nakano, Regular actions of simple algebraic groups on projective threefolds, Nagoya Math. J. 116 (1989), 139–148. MR 1029975, DOI 10.1017/S0027763000001732
Tetsuo Nakano, On equivariant completions of $3$-dimensional homogeneous spaces of $\textrm {SL}(2,\textbf {C})$, Japan. J. Math. (N.S.) 15 (1989), no. 2, 221–273. MR 1039245, DOI 10.4099/math1924.15.221
Tetsuo Nakano, Regular actions of semisimple algebraic groups on projective threefolds, especially $\textrm {SL}(2)$, Group actions and invariant theory (Montreal, PQ, 1988) CMS Conf. Proc., vol. 10, Amer. Math. Soc., Providence, RI, 1989, pp. 137–155. MR 1021286
V. L. Popov, Quasihomogeneous affine algebraic varieties of the group $\textrm {SL}(2)$, Izv. Akad. Nauk SSSR Ser. Mat 37 (1973), 792–832 (Russian). MR 0340263
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Hideyasu Sumihiro, Equivariant completion. II, J. Math. Kyoto Univ. 15 (1975), no. 3, 573–605. MR 387294, DOI 10.1215/kjm/1250523005
Hiroshi Umemura, Sur les sous-groupes algébriques primitifs du groupe de Cremona à trois variables, Nagoya Math. J. 79 (1980), 47–67 (French). MR 587409
Hiroshi Umemura, Maximal algebraic subgroups of the Cremona group of three variables. Imprimitive algebraic subgroups of exceptional type, Nagoya Math. J. 87 (1982), 59–78. MR 676586
Hiroshi Umemura, On the maximal connected algebraic subgroups of the Cremona group. II, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 349–436. MR 803342, DOI 10.2969/aspm/00610349
Hiroshi Umemura, Minimal rational threefolds. II, Nagoya Math. J. 110 (1988), 15–80. MR 945907, DOI 10.1017/S0027763000002877