On the construction of some stacks of curves and surfaces

In this thesis, we study some examples of moduli spaces in algebraic geometry, using the theory of stacks. More precisely, we study moduli stacks of curves and surfaces. We define categories whose objects are families of curves or surfaces, and we prove that these categories are stacks. In some cases, the use of algebraic spaces is required, depending on whether a natural ample line bundle exists on the objects of interest. In the context of curves, we focus on the construction of the stack of smooth genus g curves, for every integer g greater than or equal to zero. For the construction of the stack of smooth curves of genus greater than 1, we use the fact that the canonical line bundle is ample, and moreover that the tri-canonical line bundle is very ample; while for the genus 0 case we use the fact that the anti-canonical line bundle is ample. In contrast, for the case of genus 1 we use algebraic spaces in order to glue objects. In the context of surfaces, we first construct the stack of canonical models of minimal surfaces of general type. This stack does not require the use of algebraic spaces, as the canonical line bundle is ample. Then we construct the stack of minimal surfaces of general type. The latter requires the use of algebraic spaces, as in the case of curves of genus 1. We also study the process of passing from a minimal surface of general type to its canonical model, and we generalize this process for families of surfaces by constructing a morphism of stacks between the stack of minimal surfaces of general type and the stack of their canonical models.