Moduli spaces of del Pezzo surfaces of low degree

will talk about Geometric Invariant Theory constructions of moduli spaces and paramtere spaces of del Pezzo surfaces of degree 1 and 2. It is not very difficult to construct nice parameter spaces for del Pezzo surfaces, but the moduli space proves to be more challenging to find, particularly in finite characteristic. Given a parameter space with an action of a reductive group we may define stability of del Pezzo surfaces. I will talk about stability of Kahler-Einstein del Pezzo surfaces and I will discuss applications to birational geometry.