The aim of this project has been to develop a method to integrate and calculate intersection numbers on the curvilinear Hilbert scheme. Such integration on the curvilinear Hilbert scheme is interesting in its own right, but even more relevant due to a new integration technique described in [4, 5] reducing integration on larger subsets of the Hilbert scheme of points to integration on the curvilinear Hilbert scheme. These subsets include for instance the geometric subsets defined in [54], and the method provides machinery for counting hypersurfaces with prescribed singularities.

The curvilinear Hilbert scheme is highly singular, and determining the exact elements of it is a question in deformation theory of algebras. A full understanding of such deformation theory is far out of reach at the moment. Even defining the meaning of intersections on singular spaces is a diffuclt task, let alone compute such. The modern approach is often via virtual intersection theory, proving existence of so-called virtual fundamental classes allowing one to define integration. However, such virtual intersections theory lags the geometric nature of intersections as it is often not possible to give the virtual fundamental class any geometric meaning. In any case such virtual techniques are hopeless for Hilbert schemes of points on a space of large dimension.

We construct instead an explicit resolution of a birational model for the curvilinear Hilbert scheme, and use the newly developed non-reductive geometric invariant theory to calculate intersection numbers. The primary method of calculation is via equivariant localization formulas using various torus-actions, and in particular using also a localization formula in non-reductive geometric invariant theory. With these methods we are able to calculate specific intersection numbers, and in particular we state Conjecture 12.1 possibly linking the Hilbert scheme of points on surfaces to the numbers of Cayley’s formula for counting trees.

The curvilinear Hilbert scheme is an irreducible component of the punctual Hilbert scheme of points, whose points are ideals in a fixed polynomial ring. An important ingredient in the localization techniques are the torus-fixed ideals in the polynomial ring; these are exactly the monomial ideals. We use the explicit resolution of the birational model to show that all monomial ideals are contained in the same irreducible component, the curvilinear Hilbert scheme. This opens up for a new direction of further studies applying similar techniques as those applied in this work: To any monomial singularity in the punctual Hilbert scheme of points is associated a birational model as the one studied here, and via almost the same methods, one will in principle be able to obtain a complete picture of the hierarchy of monomial singularities; that is, a complete picture of the deformation theory of the algebras formed by taking the quotient with these monomial ideals.