zbMATH — the first resource for mathematics

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 \({\mathbb{Q}}[\sqrt{3}]\).

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
Full Text: DOI
[1] W. Barth andC. Peters, Automorphisms of Enriques surfaces. Invent. Math.73, 383-411 (1983). · Zbl 0518.14023 · doi:10.1007/BF01388435
[2] W.Barth, C.Peters and A.Van De Ven. Compact complex surfaces. Berlin-Heidelberg-New York 1984. · Zbl 0718.14023
[3] I. Dolgachev, On automorphisms of Enriques surfaces. Invent. Math.76, 163-177 (1984). · Zbl 0575.14036 · doi:10.1007/BF01388499
[4] V. Nikulin, Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. Algebrogeometric applications. Modern Problems of Mathematics18, 1-114 (1981). Moscow, VINITI. (Engl. Transi.: J. Soviet Math.22, 1401-1476 (1983)).
[5] I. Pjateckii-?apiro andI. ?afarevi?, A Torelli theorem for algebraic surfaces of typeK3. Izvestia AN SSSR, Ser. Math.35, 530-572 (1971). (Engl. Transl.: Math. USSR-Izvestia5, 547-588 (1971)).
[6] T.Shioda and H.Inose, On singularK3 surfaces, in Complex Analysis and Algebraic Geometry, in honor of K. Kodaira. Cambridge Univ. Press, 119-136 (1977).
[7] J. Wehler, Cyclic coverings: Deformation and Torelli theorem. Math. Ann.274, 443-472 (1986). · Zbl 0593.32017 · doi:10.1007/BF01457228
[8] J.Wehler, Hypersurfaces of the flag variety: Deformation theory and the theorems of Kodaira-Spencer, Torelli, Lefschetz, M. Noether and Serre. Preprint Munich 1986. · Zbl 0662.14029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.