Quotient Presentations of Mori Dream Spaces

Abstract

In the present thesis, we investigate quotient presentations of Mori Dream Spaces. In the first part, we show that varieties of Fano type and klt quasicones have finite iteration of Cox rings with factorial canonical master Cox ring. The variety can be presented as a quotient of the maximal spectrum of this ring by a solvable reductive group. The second part aims to present such factorial canonical rings as invariant rings of the special linear group over the complex numbers. We develop several algorithms to compute such invariants. In particular, we determine invariants in dimensions four and five for arbitrary sums of fundamental representations. Moreover, we complete the classification of complete intersection invariant rings of the special linear group. In the third part of the thesis, we classify compound du Val and canonical threefold singularities with a good two-torus action and we determine their tree of Cox ring iterations. In the last part, we give an outlook of how these different quotient presentations can possibly be combined.