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Algebraic geometry is the area of mathematics concerned with the study of systems of polynomial equations and their solution sets. We call such a solution set a variety or scheme. Instead of considering only individual systems of equations, we can also try to consider the set of all systems of equations (in a fixed number of variables) at once. This set itself can also be described by a system of equations; the corresponding geometric object is called the Hilbert scheme. Each point on the Hilbert scheme therefore corresponds to a system of equations.

In this thesis, we focus primarily on the Hilbert scheme of points, that is, we consider only the part of the Hilbert scheme that parameterises systems of equations whose solution sets consist of a finite number of points. This turns out to be a very complex object, with many irregularities, so-called singularities. In particular, the Hilbert scheme of points satisfies the algebro-geometric version of Murphy's law, meaning that every possible singularity occurs somewhere on the Hilbert scheme.

However, the Hilbert scheme also has another side: it has very good properties from the perspective of enumerative geometry. This is due to the presence of so-called virtual structures.

In this thesis, we study the Hilbert scheme and several closely related spaces from these two different perspectives. On the one hand, we show that these spaces behave even worse than expected from the perspective of singularities. On the other hand, we also find new virtual structures on these spaces and show that these actually have very good properties from the perspective of enumerative geometry.