Mori Dream Spaces and Rational Curves

森梦空间与理性曲线

• 批准号: 1001157
• 负责人: Ana-Maria Castravet
• 金额: $ 10.59万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2011-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001157&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簇几何及其模的不同方面。主要研究内容有两个：(1)稳定曲线模空间的有效且充足的锥体。这一系列项目重点关注稳定有理曲线的 Grothendieck-Knudsen 模空间。目标是研究模空间的森梦空间结构；特别是，给出其双有理收缩的模解释，并给出其总坐标环的演示。一种新的观点是将模空间解释为与称为超树的新组合结构相关的可约曲线的布里尔-诺特轨迹。 (2) 使用有理曲线的最小主导族对高等 Fano 变种进行研究。主要重点是 2-Fano 簇的分类和 Tsen 定理的推广。该项目更广泛的背景是代数几何领域，代数几何是最古老也是目前最活跃的数学分支之一，在整个数学领域有着广泛的应用，并延伸到物理和工程学。代数几何是对代数簇的研究，代数簇是由多项式方程组的零点定义的几何对象。代数簇的变化由所谓的模空间捕获，模空间本身就是具有非常丰富结构的簇。该项目旨在揭示各种曲线模空间的有趣结构（这在数学和理论物理的许多领域都是基础）。该项目影响算术和计算代数几何，这些领域在编码理论、机器人学、计算机视觉、系统发育学、统计学等方面的应用越来越多。