Singularities in Algebraic Geometry

本文讨论了两族几何、d模理论和紧闭理论中带原点奇点的不变量。像对数规范阈值或最小对数差异这样的不变量在Mori理论中扮演着重要的角色：正如Shokurov所示，它们的推测性质将对一个主要的开放问题（所谓的“翻转终止”）产生强烈的影响。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，例如伯恩斯坦多项式（来自d模理论），定义为正特征的不变量，如f纯阈值，或弧和射流空间。本提案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变量的某种分类。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图像中，而对它们的奇异性的良好理解对于分类过程至关重要。衡量奇异性好坏的不变量的一般性质在本研究中起着重要作用。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.