Cones and positivity in algebraic geometry

这个项目将讨论代数几何中的一些基本问题，代数几何是纯数学研究的一个中心领域。我们在代数几何中研究的对象是所谓的代数变分，即多项式方程组的解集。代数变体在数学的许多领域都引起了极大的兴趣，包括数论和拓扑学，但也有一系列的应用：数学物理，它们为物理对象提供数学模型；控制理论和运动规划，它们表示受代数约束的系统的可能状态；还有其他许多人。因此，对代数变量的结构有一个高层次的认识具有巨大的理论和实践意义。这个项目包括两个相互关联的方法来理解代数变量的结构问题。第一种方法是研究一个重要的假设，称为“莫里森-川俣锥猜想”。如果是真的，这个猜想将提供非常精确的信息，说明某类代数变体可以通过代数映射相互关联。我们的研究将开发一种归纳法，允许使用关于较小品种的信息来推断关于许多较大品种的信息。这将大大增加我们对这个猜想所知的范围：到目前为止，它只被证明了非常特殊类型的变种，但我们的归纳法将产生更广泛的结果。第二种方法将集中在这个问题上：给定一个代数变量，我们如何描述包含在其中的所有较小代数变量的集合？在过去的三十年里，在理解曲线（即一维变量）的问题上取得了巨大的进展，这大大提高了我们对所有代数变量的总体认识。在我们项目的这个组成部分中，我们将解决理解给定品种中所有“较大”子品种的问题——例如，各种维度4中的表面。在这个问题上的进展将产生一套新的工具来理解、区分和分类代数变量。

This project will address some basic issues in algebraic geometry, a central field of research in pure mathematics. The objects we study in algebraic geometry are so-called algebraic varieties, meaning solution sets of systems of polynomial equations. Algebraic varieties are of great interest in many parts of mathematics, including number theory and topology, but also in a range of applications: mathematical physics, where they provide mathematical models for physical objects; control theory and motion planning, where they represent the possible states of a system subject to algebraic constraints; and many others. Getting a good high-level picture of the structure of algebraic varieties is therefore of immense theoretical and practical interest.This project consists of two interrelated approaches to the problem of understanding the structure of algebraic varieties. The first approach is to investigate an important hypothesis called the "Morrison--Kawamata cone conjecture". If true, this conjecture would provide very precise information about the way in which certain classes of algebraic varieties can be related to each by algebraic mappings. Our research will develop an inductive method, allowing to use information about smaller varieties to deduce information about many larger varieties. This will greatly increase the scope of what is known about the conjecture: to date it has only been proved for very special types of varieties, but our inductive method will yield results in much greater generality.The second approach will focus on the question: given an algebraic variety, how can we describe the collection of all smaller algebraic varieties contained inside it? In the past thirty years, immense progress has been made on the problem of understanding the curves (that is, one-dimensional varieties) contained inside a given variety, and this has greatly improved our understanding of the general picture of all algebraic varieties. In this component of our project, we will tackle the problem of understanding all "larger" subvarieties in a given variety --- for example, the surfaces inside a variety of dimension four. Progress on this question will yield a new set of tools for understanding, distinguishing, and classifying algebraic varieties.