Fix an algebraically closed field k. Let Mg be the moduli space of curves of genus g over k. The main result of this note is that Mg is irreducible for every k. Of course, whether or not M s is irreducible depends only on the characteristic of k. When the characteristic is o, we can assume that k ~-(1, 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 p1 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 q-i. Unfortunately, attempts to extend this method to all p seem to get stuck on difficult questions of wild ramification. Nowadays, the Teichmtiller theory gives a thoroughly analytic but very profound insight into this irreducibility when k----C. Our approach however is closest to Severi'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 Severi's and Grothendieck's ideas is to construct families of curves X, some singular, with pa(X)-=g, over non-singular parameter spaces, which in some sense contain enough singular curves to link together any two components that Mg might have.The essential thing that makes this method work now is a recent " 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), to go from curves to abelian varieties (cf. [M2] ). Grothendieck has recently strengthened his method so that it applies in all characteristics (SGA 7, ~968) 9 Mumford has also given a proof using theta functions in char. ~2. The result is this:
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