The Hilbert Schemes of Points on Singular Varieties and Kodaira Non-Vanishing in Characteristic p

The thesis consists of two parts of work. In the first part, we study the geometry of the Hilbert schemes of points on singular curves and surfaces. In particular, the Hilbert scheme is irreducible for a surface with only cyclic quotient rational double point singularities. In the second part, we construct new examples of Kodaira non-vanishing on surfaces in positive characteristics and deduce further pathologies on their linear systems.