Log singularities, discrepancies, and thresholds with applications

DMS-0400832 Shokurov该奖项支持Shokurov教授的一个项目。所提出的研究涉及代数簇和对数对的某些奇点，包括出现在对数最小模型程序中的奇点。它们最基本的特征，如差异、阈值和互补，被研究的目的是将它们应用于4维及更高维度的LMMP。这个过程是通过一些新的和旧的猜想来阐述的。Pi Shokurov打算完成4折的对数终止，从而完成了4维的LMP，并在5维和更高维获得新的结果。可以预期的是，一些关于奇点的基本结果必须先于LMMP，但其他更高级的结果可能会交织在一起。建议澄清这种情况以及代数几何领域中已知的和新的概念、方法和猜想之间的关系，以便更好地理解LMP在其环境中的作用。重点将集中在偏差，阈值，极值射线的长度，Alexeev-Borisovs和acc型猜想，以及饱和线性系统的限制。这是代数领域中的研究方法和在代数几何中的应用。翻转或翻转是几何对象的一类重要的标准变换，其终止意味着这些变换的任何序列的有限，以及从计算角度看相应的几何算法的有效性。代数和代数几何是现代数学中非常古老、传统的领域，但在过去的几十年里，它们取得了革命性的成就。在其起源时，代数几何处理的图形可以在平面上由最简单的方程定义，即多项式，或在3-空间中由最简单的几何结构给出，例如二次曲线。代数是关于这些方程式的。这两个领域与大多数数学分支相互作用，例如分析、拓扑学和数学物理，在这些领域以及在数论、物理、离散和计算数学以及机器人学中的应用。

DMS-0400832ShokurovThis award supports a project of Professor Shokurov. The proposed research deals with certain singularities of algebraic varieties and log pairs, including ones that appear in the Log Minimal Model Program (LMMP). Their most fundamental characteristics, such as discrepancies, thresholds, and complements, are investigated with an aim to apply them to the LMMP in dimensions 4 and higher. This process is set forth by some new and old conjectures. PI Shokurov intends to finish the log termination for 4-folds, which completes the LMMP in dimension 4, and to obtain new results in dimensions 5 and higher. It is expected that some fundamental results on singularities must precede theLMMP, but other results, more advanced, could be interwoven. It is proposed to clarify this situation and the relations between known and new concepts, methods, and conjectures in the field of algebraic geometry toward better understanding of the LMMP within its environment. The focus will be on discrepancies, thresholds, lengths of extremal rays, the Alexeev-Borisovs and acc type conjectures, and on confinement of saturated linear systems.This is research in the field of algebra with methods and applications in algebraic geometry. A termination for flips or flops, an important class of standard transformation of geometrical objects, means finiteness of any sequence of those transformations, and effectiveness of corresponding geometrical algorithm from the computational point of view. Algebra and algebraic geometry are very old, traditional areas of modern mathematics, but which have had a revolutionary flowering in the past decades. In its origin, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials, or be given in the 3-space by the simplest geometric constructions, e.g., conic sections. Algebra is about these equations. Both fields interacts with most of branches of mathematics, e.g., analysis, topology and mathematical physics, with applications in those fields as well as in number theory, physics, discrete and computational mathematics, and robotics.