Notes on the Construction of the Moduli Space of Curves

Abstract

The purpose of these notes is to give an introduction to Deligne-Mumford stacks and their moduli spaces, with emphasis on the moduli problem for curves. The paper has 4 sections. In section 1 we discuss the general problem of constructing a moduli "space" of curves. In section 2 we give an introduction to Deligne-Mumford stacks and their moduli spaces. In section 3 we return to curves and outline Deligne and Mumford's proof that the stack of stable curves is a smooth and irreducible Deligne-Mumford stack which is proper over Spec Z. In section 4 we explain how to use geometric invariant to construct moduli spaces for quotient stacks. Finally, we briefly outline Gieseker's geometric invariant theory construction of the moduli scheme of projective curves defined over an algebraically closed field. These notes are a slightly revised version of notes which the author has circulated privately for several years, and are based on lectures the author gave at the Weizmann Institute in July 1994. They are also available at the URL http://math.missouri.edu/~edidin/Papers/ Any updates will be posted to this URL.