Mori dream spaces: Theory, algorithms and implementation

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Mori dream spaces are varieties with a finitely generated Cox ring. Examples are the toric varieties and the Fano varieties. Based on the fact that Mori dream spaces are quotients of affine varieties, one obtains a concrete description of them by means of algebraic-combinatorial data. This generalizes the description of toric varieties in terms of lattice fans and makes Mori dream spaces accessible for computations. We intend to study the geometry of Mori dream spaces in terms of their describing data and to develop and implement algorithms for concrete computations.