Cox rings, orbit cones and GIT-equivalence.

Let X be a normal algebraic variety with a free finitely generated
divisor class group and finitely generated total coordinate ring R(X). Then X may be realized as a good quotient of an open subset $ ilde X$ in the affine factorial variety $overline{X}:= Spec(R(X))$ by the so-called Neron-Severi torus S. The idea of the project is to lift some problems from X to $overline X$. Using this strategy, we study GIT-equivalence of G-linearized line bundles on X. Another objective is the description of equivariant embeddings with small boundary for a given homogeneous space G/H. The main combinatorial ingredient here is the GIT-fan associated with
the S-action on $overline X$ (joint project with Juergen Hausen).