Even eight on a Kummer surface.
dc.contributor.author | Mehran, Afsaneh K. | |
dc.contributor.advisor | Dolgachev, Igor | |
dc.date.accessioned | 2016-08-30T16:03:20Z | |
dc.date.available | 2016-08-30T16:03:20Z | |
dc.date.issued | 2006 | |
dc.identifier.uri | http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:3224698 | |
dc.identifier.uri | https://hdl.handle.net/2027.42/125831 | |
dc.description.abstract | This thesis addresses the problem of finding all the double covers of a given Kummer surface branched along an <italic>even eight</italic>. An even eight is defined to be a set of eight smooth disjoint rational curves whose sum determine an even class in the Picard group of the Kummer surface. By a well known result of Nikulin, this problem is equivalent to the classification of all the <italic>K</italic>3 surfaces admitting a symplectic involution such that the corresponding quotient is birational to a given Kummer surface. This dissertation is divided into two parts. In the first part, we give a complete classification of all such <italic>K</italic>3 surfaces by giving a criterion on their transcendental lattice. We show that up to isomorphism those <italic>K</italic>3 surfaces form a finite set. When the Kummer surface is general, we give a list of all the possible transcendental lattices of the corresponding <italic>K</italic>3 surfaces. The second part deals with the geometrical interpretation of some of the <italic>K</italic>3 surfaces obtained in the list above. We give new double plane models of the general Kummer surface and show that those <italic>K</italic>3 surfaces correspond to the decomposition of the branch locus of these new models. This correspondence allows us to give a full geometrical description of the Picard group of the <italic>K</italic>3 surfaces. Finally, we show that there exist in our list pairs of <italic>K</italic>3 surfaces where one is a Jacobian elliptic fibration and the other is a torsor over this Jacobian. This draws a connection between the already known classification of abelian surfaces arising as the double cover of a Kummer surface and our classification. | |
dc.format.extent | 72 p. | |
dc.language | English | |
dc.language.iso | EN | |
dc.subject | Even Eight | |
dc.subject | Kummer Surface | |
dc.title | Even eight on a Kummer surface. | |
dc.type | Thesis | |
dc.description.thesisdegreename | PhD | en_US |
dc.description.thesisdegreediscipline | Mathematics | |
dc.description.thesisdegreediscipline | Pure Sciences | |
dc.description.thesisdegreegrantor | University of Michigan, Horace H. Rackham School of Graduate Studies | |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/125831/2/3224698.pdf | |
dc.owningcollname | Dissertations and Theses (Ph.D. and Master's) |
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