PROJECTIVITY OF MODULI OF CURVES Contents

In this article, we exposit a proof, due to Kollár in [9], of the fact that the moduli stack Mg of stable curves of genus g ≥ 2 admits a coarse moduli space Mg which is projective over Z. In particular, this means that the coarse moduli space Mg, which is a priori but an algebraic space, is actually a projective scheme over Z. Together with the work of Deligne–Mumford [3], the moduli space Mg is actually an irreducible smooth projective scheme over Z.