Connectedness and persistence 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. Currently, we have obtained results on generalising Gotzmann's persistence theorem to toric varieties. We are now further looking at the connectedness of multigraded Hilbert schemes in the context of products of projective spaces.
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.