On the Model Theory of Function Fields

We study the results of Duret [6] which discuss the first-order rigidity for function fields of curves over algebraically closed fields. To do so, we apply the definability of genus, as well as definability of the field of constants, results of Duret [5]. These are the focus of Chapter 3. We show that, under certain conditions, the first order properties of a function field of a curve are enough to determine its isomorphism class.