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Minimal rational threefolds II

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura
Hiroshi Umemura*
Affiliation:
Department of Mathematics, Mie University, Tsu, 514, Japan
*
Department of Mathematics, Faculty of Science, Kumamoto University, Kumamoto, 860, Japan
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The Enriques-Fano classification ([E.F], [F]) of the maximal connected algebraic subgroups of the three variable Cremona group, despite of its group theoretic feature, seems to be the most significant result on the rational threefolds so far known. In this paper as in [MU] we interpret the Enriques-Fano classification from a geometric view point, namely the geometry of minimal rational threefolds. We explained in [MU] the link between the two objects; the maximal algebraic subgroups and the minimal rational threefolds. Let (G, X) be a maximal algebraic subgroup of three variable Cremona group. We denote by ℓ(G, X) the set of all the algebraic operations (G, Y) such that Y is non-singular and projective and such that (G, Y) is isomorphic to (G, X) as law chunks of algebraic operation: namely (G, Y) is birationally equivalent to (G, X). Then we define an order in ℓ(G, X): for (G, Z), (G, W) ∊ ℓ(G, X), (G, Z)>(G, W) if there exists an G-equivariant birational morphism of Z onto W.

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Type
Research Article
Information
Nagoya Mathematical Journal , Volume 110 , June 1988 , pp. 15 - 80
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[A] Atiyah, M. F., Complex fibre bundles and ruled surfaces, Proc. London Math. Soc. (3), 5 (1955), 407–434.Google Scholar
[Be] Beauville, A., Variété de Prym et Jacobiennes intermédiaires, Ann. Sci. Ec. Norm. Sup., 4e série, t. 10 (1977), 304–392.Google Scholar
[Bo] Borei, A., Linear algebraic groups, Benjamin, New York, 1969.Google Scholar
[Da] Danilov, V.L., Decomposition of certain birational morphisms, Math. USSR Izv., 16 (1981), 419–429.Google Scholar
[De] Demazure, M., Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. Ec. Norm. Sup., 4e série, t. 3 (1970), 507–588.Google Scholar
[EF] Enriques, F. e Fano, G., Sui gruppi di transformazioni cremoniane dello spazio, Annali di Matematica pura ed applicata, s. 2a to. 15 (1897), 59–98.Google Scholar
[F] Fano, G., I gruppi di Jonquières generalizzati, Atti della R. Acc. di Torino, 33 (1898), 221–271.Google Scholar
[Hi] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math., 79 (1964), 109–326.Google Scholar
[Hu] Humphreys, J. E., Introduction to Lie algebra and representation theory, GTM 9, Springer-Verlag, Berlin-Heidelberg-New York, 1972.Google Scholar
[I1] Iskovskih, V. A., Fano threefolds I, Math. USSR Izv., 11 (1977), 485–527.CrossRefGoogle Scholar
[I2] Iskovskih, V. A., Fano threefolds II, Math. USSR Izv., 12 (1978), 469–506.Google Scholar
[Man] Manin, Yu. I., Cubic forms, North-Holland, Amsterdam, 1974.Google Scholar
[Mar1] Maruyama, M., On classification of ruled surfaces, Lect. in Math., Dept. Math. Kyoto Univ. 3, Kinokuniya, Tokyo 1970.Google Scholar
[Mar2] Maruyama, M., On a family of algebraic vector bundles, Number Theory, Algebraic Ge ometry and Commutative Algebra, in honor of Y. Akizuki, Kinokuniya, Tokyo, 1973, 95–146.Google Scholar
[Mi] Miyanishi, M., Algebraic methods in the theory of algebraic threefolds, Advanced Studies in Pure Math., 1, 1983, Alg. Var. and An. Var., 69–99.Google Scholar
[Mo] Mori, S., Threefolds, whose canonical bundles are not numerically effective, Ann. of Math., 116 (1982), 133–176.Google Scholar
[MU] Mukai, S. and Umemura, H., Minimal rational threefolds, Algebraic geometry, Lecture Notes in Math., 1016, Springer-Verlag, Berlin-Heidelberg-New York, 1983, 490–518.Google Scholar
[Re] Reid, M., Canonical 3-folds, Journée de géométrie algébrique d’Angers, ed. Beauville, A., Sijthoff and Noordhoff, Alphen 1980, 273–310.Google Scholar
[Ro] Rosenlicht, M., Some basic theorems on algebraic groups, Amer. J. Math., 78 (1956), 401–443.Google Scholar
[Sc] Schwarzenberger, R. L. E., Vector bundles on the projective plane, Proc. London Math. Soc, (3) 11 (1961), 623–640.Google Scholar
[Su] Sumihiro, H., Equivariant completion, J. Math. Kyoto Univ., 14–1 (1974), 1–28.Google Scholar
[U1] Umemura, H., Sur les sous-groupes algébriques primitifs de groupe de Cremona à trois variables, Nagoya Math. J., 79 (1980), 47–67.Google Scholar
[U2] Umemura, H., Maximal algebraic subgroups of the Cremona group of three variables, Nagoya Math. J., 87 (1982), 59–78.Google Scholar
[U3] Umemura, H., On the maximal connected algebraic subgroups of the Cremona group I, Nagoya Math. J., 88 (1982), 213–246.Google Scholar
[U4] Umemura, H., On the maximal connected algebraic subgroups of the Cremona group II, Algebraic groups and related topics, Advanced Studies in Pure Math., 6, Kinokuniya-North-Holland, 1985, 349–436.Google Scholar
[U5] Umemura, H., Algebro-geometric problems arising from Painlevé’s works, Algebraic and Topological theories—to the memory of Dr. T. Miyata, Kinokuniya, Tokyo, 1985, 467–495.Google Scholar
[Z] Zariski, O., Introduction to the problem of minimal models in the theory of algebraic surfaces, Math. Soc. Japan, Tokyo, 1958.Google Scholar