Moduli of Varieties of General Type

自然界中出现的许多形状都是由多项式方程给出的，而计算机计算多项式的速度特别快。理解多项式方程组的解是许多领域进步的关键，包括计算机辅助几何设计、机器人技术和物理学。在大多数情况下，系数的微小变化会导致几何对象的微小变化。例如，稍微增加球体的半径会产生稍大的球体。该项目的目的是提供一个通用的理论框架来分析系数的微小变化导致几何对象发生巨大变化的情况。 在这些情况下，测量或制造中的小错误可能会产生很大的后果。该项目的主要目的是了解什么是良好的代数簇族。德利涅和芒福德构建的曲线模及其紧化是数学和弦理论中最重要的对象之一。高维模理论的目标是构造更通用的版本，并用它来理解高维代数簇的几何。 Kollar 和 Shepherd-Barron 于 1988 年提出了一种压缩一般类型簇的模空间的方法。限制对象称为稳定簇。它们满足局部条件（仅具有对数规范奇点）和全局条件（具有充足的规范类）。更一般地，按照阿列克谢耶夫，人们应该考虑射影简化格式和除数的非负线性组合（具有有理或实数系数）对的模空间。为了获得合理的模空间，这些对应该满足局部条件（该对仅具有半对数规范奇点）和全局条件（具有充足的对数规范类）。该项目的目的是应用半对数正则奇点的研究来完成稳定对的模理论。就基础品种而言，该理论是完整的，但货币对的除数部分表现出非平坦行为，用通常的技术很难接近。对于没有约数的稳定簇，模空间的存在性是已知的，但需要系统地写下来。当添加除数时，我们遇到了一个问题（首先由哈塞特观察到）：该对的变形不需要引起除数部分的平坦变形。对于半正规基空间，Chow 簇理论是理解非平坦变形的理想选择。第一个主要目标是将其推广到减少的基础空间。一般来说，当除数部分的系数大于一半时，我们可能会期望得到更好的理论。 第二个主要目标是通过将簇和除数视为本质上独立且平坦变化的对象来制定模理论。第三个目标是了解当除数部分的系数为一半时会发生什么，因为这种情况在应用中经常出现。除了其本身的重要性之外，这是一个针对人们通常可以使用的各种技术的测试用例。第四个也是最具推测性的部分是开发一种在所有情况下都能发挥最佳作用的通用模理论。一个关键问题是，有几个相互竞争的定义对某些非简化方案给出了略有不同的答案。我们需要了解精确的关系并找到最佳的理论。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。

Many shapes that occur in nature are given by polynomial equations, and computers are especially fast in computing polynomials. Understanding the solutions of systems of polynomial equations is a key to advances in many areas, including computer aided geometric design, robotics and physics. In most cases a small change of the coefficients results in a small change of the geometric objects. For example, increasing the radius of a sphere a little results in a slightly larger sphere. The aim of this project is to give a general theoretical framework for analyzing those cases when a small change of the coefficients results in a dramatic change of the geometric objects. These are the cases when a small error in measurement or manufacturing may have large consequences.The main aim of the project is to understand what a good family of algebraic varieties is. The moduli of curves and its compactification constructed by Deligne and Mumford are among the most important objects in mathematics and in string theory. The goal of higher dimensional moduli theory is to construct more general versions and to use this in understanding the geometry of higher dimensional algebraic varieties. An approach to compactifying the moduli space of varieties of general type was proposed by Kollar and Shepherd-Barron in 1988. The limiting objects are called stable varieties. These satisfy a local condition (having only log-canonical singularities) and a global condition (having ample canonical class). More generally, following Alexeev, one should consider moduli spaces of pairs of a projective, reduced scheme and a non-negative linear combination (with rational or real coefficients) of divisors. In order to get a reasonable moduli space, these pairs should satisfy a local condition (the pair has only semi-log-canonical singularities) and a global condition (having ample log canonical class). The aim of the project is to apply the study of semi-log-canonical singularities to complete the moduli theory of stable pairs. The theory is complete as far as the underlying varieties are concerned, but the divisor parts of the pairs exhibit non-flat behavior that is harder to approach with the usual techniques. For stable varieties without divisors, the existence of the moduli space are known, but it needs to be written down in a systematic way. When a divisor is added, we run into the problem (first observed by Hassett) that a deformation of the pair need not induce a flat deformation of the divisor part. For seminormal base spaces the theory of Chow varieties is ideal to understand non-flat deformations over. The first main aim is to generalize this to reduced base spaces. In general we may expect a better theory when the coefficients in the divisorial part are bigger than a half. The second main aim is to work out the moduli theory by treating the variety and the divisor as essentially independent objects varying flatly. A third aim is to understand what happens when the coefficients in the divisorial part are allowed to be a half, since this case comes up frequently in applications. Besides being important in its own right, this is a test case for the various techniques that one can use in general. The fourth, most speculative part is to develop a general moduli theory that works optimally in all cases. A key problem is that there are several competing definitions that give slightly different answers for some non-reduced schemes. We need to understand the precise relationships and to find the optimal theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.