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Some finiteness properties of the fundamental group of a smooth variety

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Some finiteness properties of the fundamental group of a smooth variety COMPOSITIO MATHEMATICA MICHAEL P. ANDERSON Some finiteness properties of the fundamental group of a smooth variety Compositio Mathematica, tome 31, no 3 (1975), p. 303-308. <http://www.numdam.org/item?id=CM_1975__31_3_303_0> © Foundation Compositio Mathematica, 1975, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 303 SOME FINITENESS PROPERTIES OF THE FUNDAMENTAL GROUP OF A SMOOTH VARIETY Michael P. Anderson COMPOSITIO MATHEMATICA, Vol. 31, Fasc. 3, 1975, pag. 303- 308 Noordhoff International Publishing Printed in the Netherlands In this paper we prove that for any smooth variety X over an algebraically closed field of characteristic p ~ 2, 3, 5 the group 03A0(p)1(X) is a finitely presented pro-(p)-group. We recall that 03A0(p)1(X) denotes the maximal quotient of 03A01(X) of order prime to p. In [8] Exposé II this result is demonstrated for smooth X provided there exists a projective smooth compactification X of X such that XBX is a divisor with normal crossings on X and for all X provided we assume strong resolution of singularities for all varieties of dimension ~ n. Thus the result was previously known for X of dimension ~ 2. The essential new step is Lemma 1 which allows us to reduce to the case of dimension 2. The proof of this lemma uses Abhyankar’s work on resolution of singularities [1] together with the technique of fibering by curves. We follow the notation of [7] Exposé XIII and [8] Exposé II. Let us now state o

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