Abstract:
The subject of this thesis are varieties with a torus action of complexity one, i.e. algebraic varieties X with an algebraic torus T acting effectively on them, where dim(T)=dim(X)-1. A combinatorial description for such varieties is provided which generalizes the convex geometrical description of toric varieties by lattice fans. The systematical construction of rational complexity-one T-varieties in terms of certain integral matrices and a collection of polyhedral cones is applied to classification problems on varieties with complexity-one torus action. In this context Fano varieties, i.e. varieties with ample anticanonical divisor are of special interest. The main focus lies on effective bounds and concrete classifications for Fano varieties with Picard number 1 and Gorenstein log del Pezzo surfaces. Due to the methods, an explicit description of the Cox rings of these varieties is obtained. Furthermore, almost homogeneous complexity-one T-varieties, i.e. varieties whose automorphism group acts with an open orbit, are discussed and combinatorially described. This combiantorial approach is used to classify log-termial almost homogeneous surfaces with Picard number 1 and only one singularity as well as three-dimensional almost homogeneous complexity-one T-varieties with reductive automorphism group.