Connectedness and persistence for multigraded Hilbert schemes

一个希尔伯特概型通过它们的希尔伯特多项式对一个投射概型的子概型进行分类。在射影空间P ^n的情况下，我们可以使用Gotzmann的两个定理，正则性和持久性，将希尔伯特方案实现为格拉斯曼方案的子方案。对于更一般的投影方案的希尔伯特计划是更复杂的，并没有一个类似的戈茨曼的持久性定理。我们的工作具体的情况下，复曲面品种，这是一个自然的下一步多级希尔伯特计划。在这种情况下，规律性是很好理解的。目前，我们已经得到的结果推广Gotzmann的持久性定理环面品种。我们现在正在进一步研究的连通性的多阶希尔伯特计划的产品的射影空间。作为一个项目在数学，这个项目属于福尔斯的职权范围内的工程和物理科学研究理事会。该项目没有正式的外部合作伙伴。

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.