CAREER: Singularities in the Minimal Model Program and Birational Geometry

本研究方向为代数几何领域。它集中讨论了分类理论中的一系列问题，这些问题是由奇点理论和最小模型程序的类似工具联系起来的。该项目有三个主要目标：(1)研究奇点数值不变量的抽象性质及其在几何中的影响，从而解决该领域长期存在的重要开放问题，如Shokurov的ACC猜想和翻转猜想的终止。(2)确定了一类射光Fano超曲面的双几何结构的刚性性质，并利用这些性质的变化引入了测量不同程度非理性的新观点。(3)研究曲线在Fano变量上的不同方面的行为，特别是在最小模型程序的范围内与变量的变形有关：如果满足预期的变形特性，将揭示Fano变量相当令人惊讶的刚性性质。其他更具推测性的项目，特别是关于Shokurov的ACC猜想与通过规范化纳什爆炸解决奇点之间可能存在的联系，以及关于Fano变种族中的Cox环的行为的项目，在整个提案中都有概述。最小模型程序，主要程序面向代数品种的两国分类，是提出的研究的总体框架。基于曲面分类模型（20世纪初意大利学派的主要成就之一）设计的最小模型程序已成为代数几何的主要趋势之一，S. Mori因其在三维变化方面的工作而获得菲尔兹奖。多年来，几位顶尖的数学家对这个程序做出了深刻的贡献，最近有了一些惊人的进展，使我们非常接近于在所有维度上完成一个完整的程序。然而，仍有许多基本问题尚未解决，其中一些问题构成了拟议研究的主要目标。本文还提出了一系列的教育活动：这些活动的范围涵盖了很大的范围，从高中生到同行研究人员，经过本科生、研究生和青年研究人员。

The proposed research is in the field of algebraic geometry. It focuses on a series of problems in classification theory that are connected by the use of similar tools coming from singularity theory and the minimal model program. The project has three main objectives:(1) Investigate abstract properties of numerical invariants of singularities and their impact in birational geometry, and thus address longstanding open problems of primary importance in the field such as Shokurov's ACC Conjectures and the Termination of Flips Conjecture. (2) Determine rigidity properties of the birational geometric structure of certain class of projective Fano hypersurfaces, and use variations of these properties to introduce a new point of view in measuring various degrees of non-rationality. (3) Study different aspects concerning the behavior of curves on Fano varieties, especially in connection to the deformations of the variety and within the context of the minimal model program: expected deformation properties, if satisfied, would bring to light a rather surprising rigidity nature of Fano varieties. Other more speculative projects, notably one regarding a possible connections between Shokurov's ACC Conjecture and resolution of singularities via normalized Nash blow-ups, and one on the behavior of the Cox ring in families of Fano varieties, are outlined throughout the proposal.The minimal model program, the main program geared towards the birational classification of algebraic varieties, is the general framework of the proposed research. Designed over the model of classification of surfaces (one of the main achievements of the Italian school at the beginning of the 20th century), the minimal model program has become one of the major trends in algebraic geometry, earning S. Mori the Fields Medal for is work on three dimensional varieties. Several leading mathematicians have deeply contributed to the program throughout the years, and very recently there has been some spectacular progress which is bringing us very close to a complete program in all dimensions. There are however many fundamental questions that still remain open, and some of these constitute the main objectives of the proposed research. A series of educational activities are also proposed: the range of such activities covers a large spectrum, from of high school students to peer researchers, passing through undergraduate students, graduate students, and young researchers.