On Fano and Calabi-Yau varieties with hypersurface Cox rings

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URI: http://hdl.handle.net/10900/118790
http://nbn-resolving.de/urn:nbn:de:bsz:21-dspace-1187907
http://dx.doi.org/10.15496/publikation-60164
Dokumentart: PhDThesis
Date: 2021-09-09
Language: English
Faculty: 7 Mathematisch-Naturwissenschaftliche Fakultät
Department: Mathematik
Advisor: Hausen, Jürgen (Prof. Dr.)
Day of Oral Examination: 2021-07-27
DDC Classifikation: 510 - Mathematics
Keywords: Algebraische Geometrie
Other Keywords: Fano-Varietät
Calabi-Yau-Varietät
Coxring
Klassifikation
Kombinatorik
Calabi-Yau variety
Cox ring
Fano variety
classification
combinatorics
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Abstract:

This thesis contributes to the explicit classification of Fano and Calabi-Yau varieties. First, we deal with complete intersections in projective toric varieties that arise from a non-degenerate system of Laurent polynomials. Here we obtain Bertini type statements on canonical and terminal singularities. This enables us to classify all non-toric terminal Fano threefolds that arise as a general complete intersection in a fake weighted projective space. The second chapter is devoted to the classification of all smooth Fano fourfolds of Picard number two that have a general hypersurface Cox ring. Using the Cox ring based description of these varieties we investigate their birational geometry and compute Hodge numbers. Moreover, we present a toolbox for constructing examples of general hypersurface Cox rings including several factoriality criteria for graded hypersurface rings. Finally, we give classification results on smooth Calabi-Yau threefolds of Picard number one and two that have a general hypersurface Cox ring.

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