Abstract
Mori dream spaces are algebraic varieties with finitely generated Cox ring; basic examples are toric varieties or rational varieties with a torus action of complexity one. Due to the finite generation of the Cox ring, Mori dream spaces allow an explicit approach in terms of commutative algebra and polyhedral combinatorics. Based on this approach we develop a series of algorithms to explore the geometry of Mori dream spaces. We first present a toolkit for basic computations with Mori dream spaces, e.g., determining the Picard group, cones of divisor classes, the canonical toric ambient variety, singularities or a test for being factorial. Specialized algorithms are presented for the case of complete intersection Cox rings or varieties with a torus action of complexity one, e.g., the computation of intersection numbers, the test for the Fano or Gorenstein properties, roots of the automorphism group, resolution of singularities. We apply these algorithms to explore and classify certain (combinatorially) minimal singular del Pezzo k*-surfaces of Picard rank two. As a first advanced algorithm, we show how to compute the Mori chamber decomposition and the GIT-fan of a torus action on an affine variety. A second series of advanced algorithms concerns the problem of the behavior of the Cox ring under modifications, e.g., of blow ups. We develop algorithms to verify finite generation, verify a guess of generators, systematically produce generators and to determine the ideal of relations of the Cox ring of the modified variety. This includes an algorithm to compute Cox rings of blow ups of Mori dream spaces that terminates if and only if the new variety is a Mori dream space. As applications, we determine the Cox rings of certain blow ups of the three-dimensional projective space and of the Gorenstein log-terminal del Pezzo surfaces of Picard number one without a non-trival k*-action. As further application, we determine the Cox rings of the smooth rational surfaces of Picard number at most six; for Picard number six, we restrict ourselves to the cases without a non-trival k*-action whereas the classification is complete for Picard number up to five.
Author and committee
dc:creator, dc:contributor.*- Author
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- Keicher, Simon
Identifiers
dc:identifier.*- Identifier
- hdl:10900/54061