Exploiting torus actions in algebraic geometry

在代数几何中利用环面作用

• 批准号: 171106430
• 负责人: Professor Dr. Klaus Altmann
• 金额: --
• 依托单位: Arbeitsgruppe Algebraische Geometrie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/171106430?language=en
• 关键词: Exploiting; torus; actions; algebraic; geometry

In many situations of algebraic geometry there exist actions of algebraic tori on objects, morphisms, or families. Algebraically, this is reflected by an (initially maybe invisible) multigrading of rank being the dimension of the torus. In the extension of this rank, this allows one to translate complicated and expensive (in terms of computing effort) algebraic geometry into algorithmically easier combinatorics and discrete/convex geometry. For many years this has been done for full torus actions (“toric varieties”). More recently, this method has also been developed and used for tori of smaller dimension. To make possible a usage of these theories in praxis, one needs the creation and implementation (in Singular) of algorithmic tools to allow free movement in a combination of algebraic and convex geometry. However, any implementation requires preparation, i.e. a further development of the computer algebra systems in question. Splendid packages exist for both convex and algebraic geometry. But none of these allow one to work with objects of both areas simultaneously.

在代数几何的许多情况下，存在代数环面对对象、态射或族的作用。在代数上，这是由一个（最初可能是不可见的）等级的多级作为环面的维度来反映的。在这个等级的扩展中，这允许将复杂和昂贵的（就计算工作而言）代数几何转换为算法上更容易的组合数学和离散/凸几何。多年来，这已经完成了完整的环面行动（“环面品种”）。最近，这种方法也被开发和用于较小尺寸的环面。为了使这些理论在实践中的使用成为可能，人们需要创建和实现（在奇异）算法工具，以允许在代数和凸几何的组合中自由运动。然而，任何实现都需要准备，即进一步开发所讨论的计算机代数系统。凸几何和代数几何都有出色的软件包。但这些都不允许同时处理这两个领域的对象。