On the birational classification of the moduli space of curves

This thesis explores the birational classification of the moduli space of stable curves of given genus over the field of complex numbers. The classification of algebraic curves represents a profound classical problem in algebraic geometry. While the moduli space of smooth curves provides a parameter space for isomorphism classes of smooth curves, it lacks properness. To apply the tools of birational geometry, it is necessary to construct a proper space by including specific singular degenerations known as stable curves, leading to the Deligne-Mumford compactification. The work is systematically structured to introduce the conceptual and technical machinery required to understand this moduli space and compute its Kodaira dimension, the most important birational invariant. The first part of the thesis recalls the notions of moduli problems and birational geometry. Subsequently, the theory of algebraic curves is developed, proceeding from smooth curves to nodal singularities, and then discussing the stability condition. In order to study the local properties of the moduli space, a chapter is devoted to deformation theory, focusing on infinitesimal deformations and the construction of the miniversal family. As the presence of curves with non-trivial automorphisms requires enlarging the algebraic category, the language of Grothendieck topologies and Deligne-Mumford stacks is thoroughly introduced. Finally, these elements merge in the study of the global properties of the moduli space of stable curves, including its properness, projectivity, and the structure of its Picard group. The ultimate focus of the thesis is the detailed computation of the Kodaira dimension carried out by Harris and Mumford in their famous article, proving that for a sufficiently large genus g, the moduli space of stable curves of genus g is a variety of general type.