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P UBLICATIONS MATHÉMATIQUES DE L ’I.H.É.S. P IERRE D ELIGNE D AVID M UMFORD The irreducibility of the space of curves of given genus Publications mathématiques de l’I.H.É.S. , tome 36 (1969), p. 75-109. < http://www.numdam.org/item?id=PMIHES_1969__36__75_0 > © Publications mathématiques de l’I.H.É.S., 1969, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www. ihes.fr/IHES/Publications/Publications.html ), implique l’accord avec les conditions générales d’utilisation ( http://www.numdam.org/legal.php ). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
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THE IRREDUCIBILITY OF THE SPACE OF CURVES OF GIVEN GENUS by P. DELIGNE and D. MUMFORD ( 1 ) Fix an algebraically closed field k. Let M.g be the moduli space of curves of genus g over k. The main result of this note is that M.g is irreducible for every k. Of course, whether or not M.g is irreducible depends only on the characteristic of k. When the characteristic is o, we can assume that k == C, and then the result is classical. A simple proof appears in Enriques-Chisini [E, vol. 3, chap. 3], based on analyzing the totality of coverings of P 1 of degree n, with a fixed number d of ordinary branch points. This method has been extended to char. p by William Fulton [F], using specializations from char. o to char. p provided that p> 2g +1. Unfortunately, attempts to extend this method to all p seem to get stuck on difficult questions of wild ramification. Nowadays, the Teichmuller theory gives a thoroughly analytic but very profound insight into this irreducibility when k=C. Our approach however is closest to Seven's incomplete proof ([Se], Anhang F; the error is on pp. 344-345 and seems to be quite basic) and follows a suggestion of Grothendieck for using the result in char. o to deduce the result in char. p. The basis of both Seven's and Grothendieck's ideas is to construct families of curves X, some singular, with ^(X)=^, over non-singular parameter spaces, which in some sense contain enough singular curves to link together any two components that M.g might have. The essential thing that makes this method work now is a recent c< stable reduction theorem " for abelian varieties. This result was first proved independently in char. o by Grothendieck, using methods of etale cohomology (private correspondence with J. Tate), and by Mumford, applying the easy half of Theorem (2.5)5 to go from curves to abelian varieties (cf. [Mg]). Grothendieck has recently strengthened his method so that it applies in all characteristics (SGA 7, 1968). Mumford has also given a proof using theta functions in char. =(=2. The result is this: Stable Reduction Theorem. Let R be a discrete valuation ring with quotient field K.
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