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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. <http://www.numdam.org/item?id=CM_1987__63_1_123_0> © Foundation Compositio Mathematica, 1987, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). 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 http://www.numdam.org/ 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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