This project studies two objects in the common ground between commutative algebra, combinatorics and algebraic geometry: the Cox rings of algebraic varieties and the Hilbert schemes of points. The Cox ring of an algebraic variety $X$ is a multigraded ring whose GIT quotients are the coordinate rings of images of $X$ via rational maps. For many interesting varieties, the so called Mori Dream spaces, the Cox ring is a finitely generated k-algebra. The project aims to develop tools to realize Cox rings of some Mori Dream spaces effectively, in terms of explicit coordinates and defining equations. Such descriptions are obtained via Koszul filtrations, via the combinatorics of Groebner bases or via new methods proposed. Specific conjectures about the structure of the Cox rings of certain classes of varieties are proposed. The project also studies the Hilbert scheme of points in affine space, in particular the problem of describing the radical component (the closure of the set of radical ideals) and the singularities near monomial ideals.
An algebraic variety is a space defined by the simultaneous vanishing of a collection of polynomial functions in several variables. A very fertile classical technique for studying the geometry of an algebraic variety is to consider the various natural coordinate systems that this variety admits. The Cox ring of an algebraic variety is a way to study all these coordinate systems simultaneously. The first objective of this project is to provide descriptions of the Cox rings of some algebraic varieties. The second objective of this proposal is to study the Hilbert scheme of points: the space of all possible configurations of n points in d-dimensional space. The geometry of this set is rather intricate and the aim of the project is to describe all possible ways in which the n-points can collide into one another and the geometry of the Hilbert scheme near these collisions.
