The research on this grant is in two areas of algebraic geometry, namely, singularities and invariant theory. In the field of singularities, the main aims are to construct geometrically meaningful compactifications of moduli spaces for surfaces of general type, to give a complete description of the versal deformation space of a quotient surface singularity and to analyze the geometry of canonical threefold singularities. In invariant theory the aim is to extend the recent results on the rationality of orbit spaces, namely, an attempt to prove the rationality of the moduli spaces of curves of genus three and five will be made. The subject of this research is algebraic geometry, the study of the geometric objects one obtains as solutions of algebraic equations. The study of the singularities in these objects (cusps etc) is important for the understanding of their properties. Moreover the rationality of space of curves of a fixed type is a topic of much research.
