Abstracts Mathematics

The Geometry of Hilbert Schemes on Projective Space

by Ritvik Ramkumar

In this thesis we study singularities of Hilbert schemes and show that there are many (components) of Hilbert schemes that are smooth or mildly singular and use them to explore phenomena in birational geometry and commutative algebra. Specifically, we study the Hilbert scheme compactification of a pair of linear spaces, describe all the sub- schemes parameterized by this component and show that it is a smooth Mori dream space. We study Hilbert schemes with two Borel-fixed points and prove that they are reduced, and that their irreducible components have normal and Cohen-Macaulay singularities. We study the Hilbert scheme of points on a threefold and extend results on the Hilbert scheme of points of a surface to this case; we also provide bounds on the dimension of this Hilbert scheme. Finally, we generalize the Hilbert and Quot schemes to construct the fiber-full scheme, which is a fine moduli space that controls all the cohomological data of a variety instead of just the Hilbert polynomial.