On Fano Arrangement Varieties

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dc.contributor.advisor Hausen, Jürgen (Prof. Dr.)
dc.contributor.author Hische, Christoff Sebastian
dc.date.accessioned 2020-06-22T10:08:35Z
dc.date.available 2020-06-22T10:08:35Z
dc.date.issued 2020-06-22
dc.identifier.other 1701742578 de_DE
dc.identifier.uri http://hdl.handle.net/10900/101709
dc.identifier.uri http://nbn-resolving.de/urn:nbn:de:bsz:21-dspace-1017093 de_DE
dc.identifier.uri http://dx.doi.org/10.15496/publikation-43088
dc.description.abstract This thesis contributes to the study of projective varieties with torus action. At first, we present an explicit approach to algebraic varieties with torus action by constructing them suitably embedded inside toric varieties. This extends existing constructions of rational varieties with a torus action of complexity one and delivers i.a. all Mori dream spaces with torus action. Our major example class are the (general/special) arrangement varieties, for which the torus action gives rise to a specific rational quotient to a projective space having as critical values a hyperplane arrangement in (general/special) position. This class comprises i.a. all toric varieties and all rational varieties with a torus action of complexity one. We present an explicit description of their Cox rings, which we use to obtain access to the geometry of these varieties, and give classification results in the smooth case. Then, we turn to singular Fano varieties. For these varieties the anticanonical complex has been introduced as a natural generalization of the toric Fano polytope and so far has been successfully used for the study of varieties with a torus action of complexity one. Using our explicit approach, we enlarge the area of application of the anticanonical complex to varieties with a torus action of higher complexity, for example, the arrangement varieties. As an application, we obtain several classification results. At first we consider intrinsinc quadrics, which are easily verified to be general arrangement varieties. We classify all Q-factorial Fano intrinsic quadrics of dimension three and Picard number one with at most canonical singularities. Moreover, considering the case of complexity two, we obtain the full classification of all Q-factorial canonical Fano intrinsic quadrics of dimension three without restrictions on the Picard number. Finally, we give classification results in the three-dimensional canonical case, where the maximal orbit quotient is the projective plane having a line arrangement of five lines in special position as its critical values. en
dc.language.iso en de_DE
dc.publisher Universität Tübingen de_DE
dc.rights ubt-podok de_DE
dc.rights.uri http://tobias-lib.uni-tuebingen.de/doku/lic_mit_pod.php?la=de de_DE
dc.rights.uri http://tobias-lib.uni-tuebingen.de/doku/lic_mit_pod.php?la=en en
dc.subject.classification Algebraische Geometrie de_DE
dc.subject.ddc 510 de_DE
dc.subject.other Cox Ringe de_DE
dc.subject.other Fano Varietäten de_DE
dc.subject.other Fano varieties en
dc.subject.other Torus actions en
dc.subject.other Torus-Wirkungen de_DE
dc.subject.other Singularities en
dc.subject.other Singularitäten de_DE
dc.subject.other Cox rings en
dc.title On Fano Arrangement Varieties en
dc.type PhDThesis de_DE
dcterms.dateAccepted 2020-05-08
utue.publikation.fachbereich Mathematik de_DE
utue.publikation.fakultaet 7 Mathematisch-Naturwissenschaftliche Fakultät de_DE

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