Construction of random points in moduli spaces and their geometry

Focusing on curves and surfaces, this project aims at constructing and studying general elements in moduli spaces of central interest. Our goal is to clarify important unirationality questions and reveal further properties of the moduli spaces and their elements. We will study Hurwitz spaces of d-sheeted coverings of the projective line by curves of genus g and numerical Godeaux surfaces, which are minimal surfaces of general type with smallest possible numerical invariants.