Exploiting torus actions in algebraic geometry

点击翻译按钮获取中文摘要

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.