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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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##### References:
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