• algebraic geometry
• moduli space
• smooth curves
• geometric invariant theory

The purpose of this thesis is to study the moduli problem for smooth projective curves. We will outline the construction of m_g: the (coarse) moduli space for smooth projective curves of genus g.
The goal is to construct a space whose "points" are in bijection with these varieties up to a chosen equivalence relation. Furthermore, we want to encode how these varieties vary continuously: for this reason we introduce the idea of family. This leads us to reformulate the problem in terms of studying the representability of a constructed functor. It turns out that, as in almost every situation, it is not representable. We could try to weaken the condition of representability by introducing the coarse moduli spaces.
Given a curve with genus ≥ 2, the very ample line bundle K^v (for v ≥ 3) embeds it in P^n. This allows us to see the curves as points of the Hilbert scheme. Roughly we consider the subscheme H_v of smooth curves that are embedded on P^n via K^v. The embedding is not unique but it depends on a choice of a basis and all different embeddings are obtained acting with PGL(n+1) on P_n. This induces an action on H_v.
We have that if the quotient of H_v by the action of PGL(n+1) is a geometric quotient (roughly an orbit space), this quotient is the coarse moduli space of m_g. For this reason, we develop geometric invariant theory (GIT) introduced by Mumford in 60' and we will apply the Hilbert-Mumford criterion following a technique due to Gieseker.