Stability of singular Fano varieties

在几何学中，正弯曲的形状，如球体，被称为法诺变种。它们的研究是数学的核心，因为它们构成了更复杂形状的基本构件。范诺品种近似于任何可以参数化的形状，因此适用于设计。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，但它们的变形可能很复杂。例如，一组圆锥形曲线可以变形成一个奇异的曲线（心形），或者两条线相交，或者一条双线。这在数学上是不可取的。法诺品种稳定性理论是近年来发展起来的，它预测了任何法诺品种族都存在一个具有良好几何性质的唯一极限。本项目旨在研究Fano品种在高维上的稳定性理论。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。下一步将是尝试证明该算法总是提供稳定极限，并最终推广到所有维度。

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.