No abstract

MATSVSAXA has proved [3] that the maximal connected group of automorphisms of a projective variety can be endowed with the structure of an algebraic group. Our aim in this note is to extend this result to arbitrary complete varieties. More generally, we shall show that a "connected" and "finite dimensional" group G of automorphisms (see below for precise definitions) of any algebraic variety X can be endowed with the structure of an algebraic group variety. The main line of argument is similar to the one used by CHEVALLv, Y [2] and SESItADRI [7] in the construction of the Picard variety, but somewhat simpler. We shall prove that the linear map of the Lie algebra of this algebraic group into the space of vector fields on X which associates to any tangent vector at the identity element of G the corresponding "infinitesimal motion" is an injection. It follows easily that G satisfies the universal property for connected algebraic families of automorphisms of X containing the identity, that is, that any algebraic family of automorphisms of an algebraic variety X parametrised by a variety T is induced by a morphism of T into G. As an application, we shall prove that the maximal connected group of automorphisms of a (locally isotrivial) principal fibre space over a complete variety has a structure of a group variety. We have been informed by the referee that this result has also been obtained by H. MATSUMVRA.All varieties will be assumed to be irreducible, and defined over an algebraically closed field K.We shall say that a family {~vt}~ of automorphisms of a variety X, where the p~ametrising set T is also a variety, is an algebraic/amily if the map T × X -~ X given by (t, x) ~ ~v~ (x) is a morphism. It is clear that if ~ : 8 -+ T is a morphism, the family {~(~)}~ s is again algebraic, and that if {~v,}~es is another algebraic family, {~v~ o ~v~}(,,~)cs× ~ is also an algebraic family. Let (J[, p) be the normatisation of X. For every t E T, ~ lifts to a unique automorphism ~vt of J[ such that p o @~ = ~ o p. We shall show that {@~}~T is an algebraic family of automorphisms of J[. Let (~, q) be the normalisation of T. Then (~ × J[, q x p) is the normalisation of T × X, and the morphism T×X£X lifts to a unique morphism @:~× J[-~ such that po = ¢ o (q × p). It follows that for any ~ E ~, the morphism of J~ onto itself * The author wishes to express his gratitude to Professors M. S. NAI~S~m~N and C. S. SESHADBI for many helpful suggestions and discussions.

No abstract

No abstract

Davenport has proved (3) that any cubic form in 32 or more variables with rational coefficients has a non-trivial rational zero. He has also announced that he has subsequently been able to reduce the number of variables to 29. Following the method of (3), we shall prove that any cubic form over any algebraic number field has a non-trivial zero in that field, provided that the number of variables is at least 54. The following is the precise form of our result.