Stability of singular Fano varieties

Fano 单一品种的稳定性

• 批准号: 2883717
• 金额: --
• 依托单位: University of Nottingham
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2883717
• 关键词: Stability; singular; Fano; varieties

In geometry, shapes that are positively curved, such as sphere, are called Fano varieties. Their study is central in mathematics as they form basic building blocks of more complicated shapes. Fano varieties approximate any shape that can be parametrised, hence are applicable in design. However, their primary application is within pure mathematics as well as theoretical physics. Although Fano varieties have simple geometry, their deformation could be complicated. For instance, a family of circular conics can deform into a singular conic (heart shape), or into two lines crossing, or a double line. This is mathematically undesired. The theory of stability for Fano varieties has been recently developed and predicts the existence of a unique limit with good geometric properties for any family of Fano varieties. This project aims to examine the stability theory for Fano varieties in higher dimensions. In first steps, several examples will be studied to predict a general algorithm to compute the stable models. The next step would be an attempt to prove that the algorithm always provides the stable limit, and finally to generalise to all dimensions.

在几何学中，正弯曲的形状，如球体，被称为Fano簇。它们的研究是数学的核心，因为它们形成了更复杂形状的基本构建块。Fano类近似于任何可以参数化的形状，因此在设计中是适用的。然而，它们的主要应用是在纯数学和理论物理中。虽然Fano变种具有简单的几何形状，但它们的变形可能是复杂的。例如，一个圆锥曲线族可以变形为一个奇异的圆锥曲线（心形），或者两条相交的直线，或者一条双线。这在数学上是不期望的。Fano簇的稳定性理论是最近发展起来的，它预言了任何Fano簇族都存在唯一的具有良好几何性质的极限。本计画的目的是研究高维空间中Fano簇的稳定性理论。在第一步中，将研究几个例子来预测计算稳定模型的通用算法。下一步将是试图证明该算法总是提供稳定的限制，并最终推广到所有维度。