Projective embeddings of algebraic varieties

During my DPhil studies in Oxford, I will be working in the field of Algebraic Geometry. This field of mathematics studies properties of objects called algebraic varieties: geometric shapes defined by polynomial equations. For instance, the equations y = x2 and y2 = x2 both come from second degree polynomials y - x2 and y 2 - x2 , yet they describe very different geometric objects in the plane: the former yields a classic 'U' shaped graph (an irreducible variety, the parabola, defined by an irreducible polynomial), while the latter gives an 'X' shape (a reducible union of two lines with equations y = x and y = -x , obtained by factoring the defining polynomial). Reducibility or otherwise is just one example of a property we can assign to a given algebraic variety in order to classify them.
Many naturally occurring objects in mathematics can be described by a set of polynomials, but the number of variables and equations involved may be very large. The resulting geometric objects can be difficult to describe explicitly. However, we can use methods in algebraic geometry, aided by computer algebra packages such as Macaulay2 or Singular, to establish properties of these objects: their dimension, being singular or otherwise, and others. Popular examples include sets of linear subspaces of given dimension of some vector space (Grassmannian and flag varieties), or the moduli space of curves of genus g with some number of marked points, which I studied in an earlier summer project.
The ubiquitous nature of algebraic sets across mathematics gives algebraic geometry a broad array of applications. In robotics, the equations defining movement of different parts relative to one another boils down to a system of polynomial equations. Modern cryptography uses elliptic curves, which are examples of algebraic varieties. Algebraic geometry is also of interest to string theorists in theoretical physics, where space itself is considered as 10-dimensional, with a 6-dimensional 'component' a special kind of algebraic variety.
My work will aim to understand projective embeddings of algebraic varieties in some specific contexts, using the Graded Ring method pioneered by Miles Reid, combined with computer algebra methods. Moduli spaces of curves mentioned before, as well as certain moduli spaces attached to algebraic surfaces called Hilbert schemes of points, will be a particular focus of interest.
This project falls within the EPSRC Research Area Algebra, and will be supervised by Professor Balazs Szendroi.