Quotient Presentations of Mori Dream Spaces

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URI: http://hdl.handle.net/10900/97666
Dokumentart: PhDThesis
Date: 2020-02-06
Language: English
Faculty: 7 Mathematisch-Naturwissenschaftliche Fakultät
Department: Mathematik
Advisor: Hausen, Jürgen (Prof. Dr.)
Day of Oral Examination: 2019-11-08
DDC Classifikation: 510 - Mathematics
Keywords: Algebraische Geometrie , Birationale Geometrie , Quotient , Invariantentheorie
Other Keywords: Cox-Ring
Iteration von Cox-Ringen
Varietät vom Fano Typ
klt Singularität
kanonische Singularität
vollständiger Durchschnitt
Spezielle lineare Gruppe
Cox ring
iteration of Cox rings
variety of Fano type
klt singularity
canonical singularity
complete intersection
special linear group
ring of invariants
Mori Dream Space
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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.

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