Log singularities, discrepancies, and thresholds with applications

该奖项支持Shokurov教授的一个项目。提出的研究涉及代数变量和对数对的某些奇异性，包括出现在对数最小模型程序（LMMP）中的奇异性。研究了它们最基本的特征，如差异、阈值和补足，目的是将它们应用于维度4和更高的lmpp。这个过程是由一些新的和旧的猜想提出的。PI Shokurov打算完成4倍的对数终止，完成第4维的LMMP，并在第5维及更高维度上获得新的结果。预计在mmp之前必须有一些关于奇点的基本结果，但其他更先进的结果可能会交织在一起。为了更好地理解lmpp的环境，我们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。重点将放在差异、阈值、极端射线的长度、阿列克谢耶夫-鲍里索夫猜想和acc型猜想，以及饱和线性系统的约束。这是在代数领域的研究与代数几何的方法和应用。翻转变换是一类重要的几何对象的标准变换，它的终止意味着这些变换序列的有限性，以及相应的几何算法从计算的角度来看的有效性。代数和代数几何是现代数学中非常古老的传统领域，但在过去的几十年里，它们出现了革命性的发展。在其起源中，代数几何处理的图形可以在平面上由最简单的方程（即多项式）定义，或者在三维空间中由最简单的几何结构（如圆锥截面）给出。代数就是关于这些方程的。这两个领域都与数学的大多数分支相互作用，例如分析、拓扑和数学物理，并在这些领域以及数论、物理学、离散和计算数学和机器人技术中应用。

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.