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On the birational automorphism groups of algebraic varieties

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On the birational automorphism groups of algebraic varieties COMPOSITIO MATHEMATICA MASAKIHANAMURA On the birational automorphism groups of algebraic varieties Compositio Mathematica, tome 63, no 1 (1987), p. 123-142. <> © Foundation Compositio Mathematica, 1987, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les conditions gé- nérales d’utilisation ( Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 123 On the birational automorphism groups of algebraic varieties MASAKI HANAMURA Department of Mathematics, Faculty of Science, Osaka University, Toyonaka, Osaka, 560 Japan Received 27 June 1986 and 5 January 1987; accepted 9 February 1987 Compositio Mathematica 63: 123-142 (1987) © Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands Introduction Let X be a projective variety over an algebraically closed field of charac- teristic zero. The purpose of this paper is to define the scheme Bir (X) of birational automorphisms of X and study its structure. It was Weil [14] who introduced the notion of birational action of an algebraic group on a variety. Many authors up to today have worked on such algebraic groups (see Rosenlicht [12], for example). On the other hand, since the construction of Hilbert schemes due to Grothendieck [2], a fruitful general philosophy has been established in the study of certain algebraic objects which appear in algebraic geometry - subschemes of a given scheme, vector bundles on a given variety, all curves of fixed genus, polarized varieties, etc. This suggests first to construct the universal

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