Compactifications of Hilbert schemes of complete intersections via Non-Reductive Geometric Invariant Theory

A complete intersection is a special kind of geometric space. It is the zero locus of a set of polynomial equations in a number of variables, and its dimension is determined by the number of variables and the number of equations. Like most geometric spaces, complete intersections come in continuous families: they can be deformed, that is modified very slightly, without changing the property of being a complete intersection. A Hilbert scheme of complete intersections is a parameter space which describes all complete intersections of the same type and the kind of continuous families they form. Hilbert schemes of complete intersections are usually not compact, which means that complete intersections can be deformed into spaces which are not complete intersections. For various applications it is useful to find a natural way to compactify such Hilbert schemes. This can be understood as giving a description of objects into which complete intersections can be modified. The aim of this project is to construct such compactifications using novel techniques of Non-Reductive Geometric Invariant Theory (GIT). GIT allows algebraic geometers to construct new spaces from simpler spaces by getting rid of their unnecessary symmetries. Recently developed Non-Reductive GIT allows us to deal with more complicated and intricate symmetries.
This project falls within the EPSRC Geometry and Topology research area. No companies or collaborators are involved.