Singularities in Algebraic Geometry

该提案涉及奇点的不变量与双有理几何，D-模理论和紧闭包理论的起源。不变量，如对数正则阈值或最小对数差异发挥重要作用的森理论：如所示Shokurov，其结构属性将有很强的影响的一个主要的开放问题（所谓的终止翻转）。另一方面，人们已经认识到这些不变量与其他领域的不变量密切相关，例如与伯恩斯坦多项式（来自D-模理论），与定义在正特征中的不变量，如F-纯阈值，或与弧和喷流空间。这个建议的主要目标是进一步研究这些不同方法之间的联系，并将其应用于更好地理解这些不变量。代数几何的长期目标之一是给出非奇异簇的某种分类。在过去的25年里，一个基本的见解是奇异空间自然地出现在画面中，并且很好地理解它们的奇异性对于分类过程至关重要。测量奇点有多坏的不变量的一般性质在这项研究中起着重要的作用。我们的计划是从不同的角度使用方法来阐明这些一般属性。

The proposal deals with invariants of singularities with the origin in birational geometry, D-module theory and tight closure theory.Invariants like the log canonical threshold or minimal log discrepancies play an important role in Mori Theory: as shown by Shokurov, their conjectural properties would have strong implications to one of the main open problems (the so-called Termination of Flips). On the other hand, it has been realized that these invariants are closely related with invariants from other fields, for example with the Bernstein polynomial (from D-module theory), with invariants defined in positive characteristic, such as the F-pure threshold, or with spaces of arcs and jets. The main goal of this proposal is to further study the connections between these different approaches, and to apply this towards a better understanding of these invariants.One of the long-term goals in algebraic geometry is to give some sort of classification of nonsingular varieties. A basic insight in the last twenty-five years is that singular spaces appear naturally into the picture, and that a good understanding of their singularities is crucial for the classification process. General properties of the invariants that measure how bad the singularities are play an important role in this study. The plan is to use approaches from various perspectives to shed some light on these general properties.