Projectivity of the moduli space of stable curves

Abstract

In this chapter we establish the projectivity of the moduli schemes \(\overline{M}_{g,n}\). To prove this we use a mixture of two techniques that are of independent interest. The first one is Mumford’s geometric invariant theory. We prove the Hilbert–Mumford criterion of stability, and we use the criterion to prove the stability of the ν-log-canonically embedded smooth curves, viewed as points in the appropriate Hilbert scheme. We then use stability of smooth curves to find numerical inequalities among cycles in moduli spaces and, consequently, positivity results. Using the same techniques, we then prove the ampleness of Mumford’s class κ 1, and hence the projectivity of \(\overline{M}_{g,n}\).