Connectedness and persistence for multigraded Hilbert schemes

Hilbert格式通过其Hilbert多项式对射影格式的子格式进行分类。在射影空间P^n的情况下，我们可以利用两个戈兹曼定理，正则性定理和持久性定理，将希尔伯特格式作为格拉斯曼格式的子格式来实现。对于更一般的射影格式，希尔伯特格式是更复杂的，并且没有类似于戈兹曼的持续定理。我们专门研究环型的多阶希尔伯特格式，这是很自然的下一步。在这种情况下，规律是很容易理解的。目前，我们已经得到了将Gotzmann的持久性定理推广到环变的结果。我们现在进一步研究在射影空间积的情况下多重梯度希尔伯特格式的连通性。作为一个数学项目，这个项目属于工程与物理科学研究委员会的职权范围。这个项目没有官方的外部合作伙伴。

A Hilbert scheme classifies subschemes of a projective scheme by their Hilbertpolynomial. 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.