Even eight on a Kummer surface.

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.