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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.

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

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