Structural properties of Cox rings of T-varieties

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Dokumentart: PhDThesis
Date: 2018
Language: English
Faculty: 7 Mathematisch-Naturwissenschaftliche Fakultät
Department: Mathematik
Advisor: Hausen, Jürgen (Prof. Dr.)
Day of Oral Examination: 2018-03-16
DDC Classifikation: 510 - Mathematics
Keywords: Algebraische Geometrie , Torische Varietät , Algebraische Varietät , Geometrische Invariantentheorie
Other Keywords: Torus-Wirkung
Iteration von Cox-Ringen
log terminal
höhere Komplexität
Cox ring
torus action
iteration of Cox rings
higher complexity
Fano variety
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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.

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