Exploiting torus actions in algebraic geometry

在代数几何的许多情形中，都存在代数环面对对象、态射或族的作用。在代数上，这反映在一个(最初可能看不见的)等级的多重等级是环面的维度。在这个等级的扩展中，这允许一个人将复杂和昂贵的代数几何转化为算法上更容易的组合学和离散/凸几何。多年来，这一直是针对完整的圆环动作(“圆环变化”)所做的。最近，这种方法也被发展并用于较小尺寸的环面。为了能够在实践中使用这些理论，人们需要创建和实现(单数)算法工具，以允许在代数几何和凸几何的组合中自由移动。然而，任何实施都需要准备，即进一步发展有关的计算机代数系统。无论是凸几何还是代数几何，都有很好的工具包。但所有这些都不允许人们同时处理这两个区域的对象。

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.