Cox's construction for $T$-varieties of complexity one and applications

Beside the description via cones and fans there is a well known quotient construction for toric varieties due to Cox, which leads to “total coordinates” on the toric variety. This contruction can be generalized to a broader class of varieties by introducing the total coordinate ring or Cox ring, respectively, of an algebraic variety. Aim of the talk is to study this construction for the case of a variety with torus action of complexity one. We use polyhedral complexes associated to points on $\mathbb P^1$ to describe such varieties and similar to the toric case we can pass from this decription to the quotient construction via Cox rings.