Local picture of twisted curves: with an Introduction to the Theory of Deligne-Mumford Stacks

A nodal curve C over a field is, roughly speaking, a curve whose singular points are ordinary double points. The classical way to describe families of such curves over a base scheme S is through a flat morphism f : C → S, whose fibers are nodal curves over the residue field. It is possible to endow nodal curves, over a base scheme S, with the action of a group of roots of unity. A twisted curve over a base scheme is a nodal curve which acquires an orbifold structure at its nodes, through the action of the roots of unity. The goal of this thesis is to prove a characterization of twisted curves over a base scheme in terms of its geometric fibers.