Mori Dream Spaces and Rational Curves

森梦空间与理性曲线

• 批准号: 1160626
• 负责人: Ana-Maria Castravet
• 金额: $ 6.27万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1160626&HistoricalAwards=false
• 关键词: Mori; Dream; Spaces; Rational; Curves

The project aims at understanding different aspects of the geometry of algebraic varieties and their moduli. There are two main topics:(1) Effective and ample cones of moduli spaces of stable curves. This sequence of projects is focused on the Grothendieck-Knudsen moduli space of stable rational curves. The goal is to investigate the Mori Dream Space structure of the moduli space; in particular, give modular interpretations for its birational contractions and give a presentation for its total coordinate ring. A new point of view is the interpretation of the moduli space as a Brill-Noether locus of a reducible curve associated to new combinatorial structures called hypertrees. (2) A study of higher Fano varieties using minimal dominating families of rational curves. The main focus is on the classification of 2-Fano varieties and generalizations of Tsen's theorem.The broader context of the project is the area of algebraic geometry, one of the oldest and currently one of the most active branches of mathematics, with widespread applications throughout mathematics and reaching into physics and engineering. Algebraic geometry is the study of algebraic varieties, which are geometric objects defined by the zeros of systems of polynomial equations. The variation of algebraic varieties is captured by the so-called moduli spaces, which are themselves varieties with a very rich structure. The project aims at revealing the intriguing structure of various moduli spaces of curves (which are fundamental in many areas of mathematics and in theoretical physics). The project impacts arithmetic and computational algebraic geometry, areas which have increasing applications in coding theory, robotics, computer vision, phylogenetics, statistics, etc.

该项目旨在了解代数品种及其模量的几何形状的不同方面。有两个主要主题：（1）稳定曲线的模量空间的有效和充足的锥。这一系列项目集中在稳定理性曲线的Grothendieck-Knudsen模量空间上。目的是研究模量空间的莫里梦境结构。特别是，给出模块化解释其生育收缩，并为其总坐标环提供了介绍。一个新的观点是将模量空间解释为与称为Hyperrees的新组合结构相关的可还原曲线的光合度轨迹。 （2）使用最小的有理曲线统治家族对较高的Fano品种的研究。主要的重点是对Tsen定理的2-Fano品种的分类和概括。该项目的更广泛背景是代数几何的领域，代数是最古老，目前最活跃的数学分支之一，它是整个整个应用程序的广泛应用程序之一数学并进入物理和工程。代数几何形状是对代数品种的研究，它们是由多项式方程系统的零定义的几何对象。代数品种的变化被所谓的模量空间捕获，它们本身就是具有非常丰富的结构的品种。该项目旨在揭示各种曲线模量空间的有趣结构（在许多数学领域和理论物理学领域都是基本的）。该项目影响算术和计算代数几何形状，这些几何形状在编码理论，机器人技术，计算机视觉，系统发育，统计等方面具有越来越多的应用。