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K3-surfaces with Picard number 2. (English) Zbl 0602.14038
The general K 3-surface X in the 3-dimensional flag variety \({\mathbb{F}}\) of \({\mathbb{P}}^ 2\) comes with two 2-fold coverings \(X\to {\mathbb{P}}^ 2\) branched along two sextic curves. Starting from the M. Noether theorem which says \(Pic(X)={\mathbb{Z}}^ 2\) we determine by elementary methods the group of automorphism as \(Aut(X)={\mathbb{Z}}_ 2*{\mathbb{Z}}_ 2\), the free product of two cyclic groups of \(rank^ 2.\) Each factor is generated by the Galois transformation belonging to one of the two coverings.
The proof makes use of the faithful representation \(Aut(X)\hookrightarrow O^ +(Pic(X))\) in the orthochronous Lorentz group of the hyperbolic lattice Pic(X), and the latter group can be computed with the help of Dirichlet’s unit theorem for the number field \({\mathbb{Q}}[\sqrt{3}]\).

MSC:
14J25 Special surfaces
11R11 Quadratic extensions
14L30 Group actions on varieties or schemes (quotients)
14J28 \(K3\) surfaces and Enriques surfaces
14J50 Automorphisms of surfaces and higher-dimensional varieties
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