Gotzmann's persistence theorem for multigraded Hilbert schemes

A Hilbert scheme classifies subschemes of a projective scheme by their Hilbert
polynomial. In the case of projective space P
n we can use two theorems of Gotzmann, regularity and persistence, to realise the Hilbert scheme as a subscheme
of a Grassmannian. For more general projective schemes the Hilbert scheme is
more complex, and there is not an analogue of Gotzmann's persistence theorem. We work specifically on the case of multigraded Hilbert schemes of toric
varieties, which is a natural next step. In this case regularity is well understood.
An overarching aim is to find a corresponding persistence theorem for multigraded Hilbert schemes, thus allowing us to realise these Hilbert schemes as
subschemes of a Grassmannian. One concrete objective is to better understand
the Hilbert scheme of a toric variety whose Picard group has rank 2. Kleinschmidt's classification allows us to understand concretely what the Cox ring
for such a toric variety looks like.
As a project in mathematics, this project falls under the remit of the Engineering and Physical Sciences Research Council. There are no official external
partners for this project.