Divisor class groups of rational trinomial varieties

We give an explicit description of the divisor class groups of rational trinomial varieties. As an application, we relate the iteration of Cox rings of any rational variety with torus action of complexity one to that of a Du Val surface.Comment: 17 page

Structural properties of Cox rings of T-varieties

In the present thesis we generalize the Cox ring based description for complete rational varieties with torus action of complexity one to Mori dream spaces with effective torus action of arbitrary high complexity. In the first part of this thesis we complete the picture for varieties of complexity one by treating the non complete, e.g affine, case. With this approach to affine varieties with torus action of complexity one, we characterize iterability of Cox rings in numerical terms. This enables us to regard log terminal singularities of arbitrary dimension with torus action of complexity one, in a larger sense, as quotient singularities, comparable to the well known surface case. In the second part, we present a constructive approach to Cox rings of Mori dream spaces with a torus action of arbitrary complexity. We study a sample class comprising the complexity one case, the so called arrangement varieties, and give concrete classification results for Fano arrangement varieties of Picard number one and for Fano arrangement varieties of complexity and Picard number two

Towards classifying toric degenerations of cubic surfaces

We investigate the class of degenerations of smooth cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. More precisely, we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and who leave the question of classifying these degenerations in the degree 3 case as an open problem. In order to carry out this classification we describe an approach which is closely related to tropical geometry and present partial results in this direction.Comment: v2: 21 pages, section 1 rewritten, added sections 6 and

Towards classifying toric degenerations of cubic surfaces

We investigate the class of degenerations of smooth cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. More precisely, we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and who leave the question of classifying these degenerations in the degree 3 case as an open problem. In order to carry out this classification we describe an approach which is closely related to tropical geometry and present partial results in this direction. &nbsp