Maximal Algebraic Subgroups of the Cremona Group of Three Variables Imprimitive Algebraic Subgroups of Exceptional Type
نویسندگان
چکیده
Enriques and Fano [4], [5] classified all the maximal connected algebraic subgroups of Cr3. Our aim is to give modern and rigorous proofs to their results. In [10], we studied the primitive subgroups. In this paper, we deal with exceptional imprimitive groups. The imprimitivity is an analytic notion. The natural translation of the imprimitivity in algebraic geometry is the de Jonquieres type operation (definition (2.1)). Every de Jonquieres type operation is imprimitive. However, the difference of these notions is subtle. The imprimitive algebraic operations in Cr3 which are not of de Jonquieres type are rather exceptional; there are only 3 such operations (theorem (3.26)). This paper together with [10] recovers all the results on Cr3 of Enriques and Fano [4]. It remains only to reconstruct Fano's classification [5] of the de Jonquieres type operations, which shall be done in our forthcoming paper. Our technique is rather old; the classification of 3 dimensional primitive operations, a very easy part of invariant theory which are of 19th century, combined with the theory of algebraic groups and transformation spaces of A. Weil. As the 4 dimensional primitive law chunks of analytic operations are classified, our method can be applied to the 4 dimensional Cremona group Cr4. We use the notations and the conventions of [10]. Therefore all manifolds, analytic groups, algebraic varieties, etc. are defined over C. The transformation spaces X of analytic or algebraic law chunk or operation (G, X) are connected. However, a differences lies in the language that we employ. Here is our French-English dictionary:
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