On the projectivity of the moduli spaces of curves.

Abstract

1. There are basically three algebraic proofs of the projectivity of the moduli spaces of stable curves: the original one by Knudsen [8][9], the proof by Mumford [12] and Gieseker [5], which makes heavy use of the machinery of geometric invariant theory, and the more recent one by Viehweg [15] and Kollar [10], which relies instead on the semipositivity of the direct images of powers of the relative dualizing sheaf and, at least in Kollar’s version, does not use geometric invariant theory at all. In this note, combining ideas from the above papers and from [4], we shall outline a proof of projectivity which, we believe, is simpler and more direct than any of the existing ones. However, while the proofs by Mumford, Gieseker, Viehweg, and Kollar are applicable, at least in principle, to a wide variety of moduli problems, ours uses in an essential way the peculiarities of the problem at hand, and it is hard to see how it could be extended to other situations. Mumford’s idea is to use the machinery of geometric invariant theory to directly construct the moduli space of stable curves as a projective quotient of the Chow or Hilbert scheme of pluricanonically embedded stable curves by the action of the projective linear group. To be able to do so, one needs to show that the k-canonical images of stable curves are stable in the invariant-theoretic sense for high k. This is quite well understood, although not really easy, for smooth curves; one notices that k-canonically embedded stable curves are linearly stable, and then proves that, for smooth curves, linear stability implies invariant-theoretic stability. In the singular case, by contrast, both Mumford and Gieseker have to rely on indirect arguments which are quite long and involved. Our main point is that one can avoid proving the invariant-theoretic stability of singular stable curves provided one is willing to grant that moduli space exists as a complete algebraic space; that this is the case, incidentally, is relatively easy to prove. As in [10], we prove projectivity by applying to a suitable invertible sheaf one of the standard numerical ampleness criteria (Seshadri’s criterion in our case). The necessary numerical estimates are obtained using the techniques of [4]. More precisely, the invariant-theoretic stability of linearly stable smooth curves is used to prove an inequality for families of stable curves which, suitably interpreted, says that the hypothesis of Seshadri’s criterion is satisfied for curves in moduli which are not contained in the boundary. The case of curves lying in the boundary is then reduced to the previous one by standard elementary techniques; in order to properly carry out this procedure, which can be viewed as a sort of induction on the genus, it is convenient to deal simultaneously with the moduli spaces of stable n-pointed curves as well. In a sense, then, our approach is to use invariant-theoretic stability as a means of obtaining numerical inequalities, rather than as a step in the construction of quotients by group actions.