Singularities in Algebraic Geometry

该提议涉及起源于二次几何、D-模理论和紧闭包理论的奇点的不变量。不变量如对数正则阈值或最小对数偏差在Mori理论中起着重要的作用：正如Shokurov所表明的，它们的猜想性质将对一个主要的公开问题(所谓的翻转的终止)具有很强的影响。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，例如与Bernstein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)、或与弧线和喷流的空间密切相关。这一建议的主要目的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，奇异空间自然而然地出现在画面中，而且对它们的奇点的良好理解对于分类过程至关重要。衡量奇点有多糟糕的不变量的一般性质在这项研究中扮演着重要的角色。我们的计划是从不同的角度使用方法来阐明这些一般性质。

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.