The Moduli Space of Curves

In this section, we will give a sketch of the construction of the moduli space Mg of curves of genus g and the closely related moduli space Mg,n of n-pointed curves of genus g using two different approaches. Throughout this section we always assume that 2g − 2 + n > 0. In the first approach, we will embed curves in a projective space P using a sufficiently high power n of their dualizing sheaf. Then a locus in the Hilbert scheme parameterizes n-canonically embedded curves of genus g. The automorphism group PGL(N + 1) acts on the Hilbert scheme. Using Mumford’s Geometric Invariant Theory, one can take the quotient of the appropriate locus in the Hilbert scheme by the action of PGL(N + 1) to construct Mg. In the second approach, one first constructs the moduli space of curves as a Deligne-Mumford stack. One then exhibits an ample line bundle on the coarse moduli scheme of this stack.