Singularities and virtual structures on the Hilbert scheme of points and its variants

Abstract

This thesis concerns the study of the Hilbert and Quot schemes of points on affine spaces A^d, and several other moduli spaces that are closely related. There is a focus on both the singularities of these moduli spaces, and on the virtual structures that these spaces admit which give rise to certain enumerative invariants. The thesis consists of several chapters studying different aspects of this overarching theme. Chapter 2 studies the singularities of a certain torus fixed locus of the Quot scheme of points on A^d. The torus considered is the d-dimensional torus which acts via its standard action on A^d. It is in particular shown that when d is at least 4, then this torus fixed locus satisfies Murphy's law, meaning that it is highly singular. Chapter 3 considers the Behrend function on the Hilbert scheme of points on A^3. It is shown that this function is not constant, disproving a conjecture stating the contrary. Chapter 4 studies certain moduli spaces parametrising quiver representations, stemming from four dimensional quivers with superpotential. These moduli spaces can be seen as a noncommutative analogue of the Quot scheme of points on A^4. We show that these moduli spaces admit virtual structures similar to the virtual structure on the Quot scheme, and study the resulting invariants. Chapter 5 studies the enumerative geometry of the Iarrobino scheme. This is a recently introduced moduli space with close ties to the (oriented) Gorenstein locus of the Hilbert scheme, and the variety of completed quadrics. Its enumerative geometry exhibits strong links to the enumerative geometry of the Hilbert scheme.