Notes on the construction of the moduli space of curves

The purpose of these notes is to discuss the problem of moduli for curves of genus g ≥ 3 1 and outline the construction of the (coarse) moduli scheme of stable curves due to Gieseker. The notes are broken into 4 parts. In Section 1 we discuss the general problem of constructing a moduli “space” of curves. We will also state results about its properties, some of which will be discussed in the sequel. We begin Section 2 by recalling from [DM] (see also [Vi]) the definition of a groupoid, and define the moduli groupoid of curves, as well as the quotient groupoid of a scheme by a group. We then discuss morphisms and fiber products in the 2-category of groupoids. Once this is in place we state the geometric conditions required for a groupoid to be a stack, and prove that the quotient groupoid of a scheme by a group is a stack. After discussing properties of morphisms of stacks, we define a Deligne-Mumford stack and prove that if a group acts on a scheme so that the stabilizers of geometric points are finite and reduced then the quotient stack is Deligne-Mumford. In the last part of Section 2 we talk about some basic algebro-geometric properties of Deligne-Mumford stacks. In Section 3 the notion of a stable curve is introduced, and we define the groupoid of stable curves. The groupoid of smooth curves is a subgroupoid. We then prove that the groupoid of stable curves of genus g ≥ 3 is equivalent to the quotient groupoid of a Hilbert scheme by the action of the projective linear group with finite, reduced stabilizers at geometric points. By the results of the previous section we can conclude the the groupoid of stable curves is a Deligne-Mumford stack defined over SpecZ (as is the groupoid of smooth curves). We also discuss the results of [DM] on the irreducibility of the moduli stack. We begin Section 4 by defining the moduli space of a Deligne-Mumford stack, proving that a geometric quotient of a scheme by a group is the