Exploiting torus actions in algebraic geometry

在代数几何的许多情况下，存在着代数环面对物体、态射或族的作用。在代数上，这反映在一个（最初可能是不可见的）秩的乘法是环面的维数。在这个等级的扩展中，它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几何和离散/凸几何。多年来，这已经完成了全环体行动（“环体品种”）。最近，该方法也得到了发展，并用于较小尺寸的环面。为了在实践中使用这些理论，人们需要创建和实现（以单数形式）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，任何实现都需要准备，即对所讨论的计算机代数系统进行进一步的开发。对于凸几何和代数几何，都存在出色的包。但这些都不允许一个人同时处理两个领域的对象。

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.